Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem? The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective on points in the sense that the induced map $\text{Hom}(1, X) \to \text{Hom}(1, Y^X)$ is surjective, then $Y$ has the fixed point property: for every morphism $g : Y \to Y$ there exists a point $y : 1 \to Y$ such that $g \circ y = y$. 
The Brouwer fixed point theorem asserts that the closed $n$-disks, all of which I will denote by $D$ for ease of notation, have the fixed point property as objects of $\text{Top}$. 
Seeing these two theorems together, it is tempting to try to prove the latter from the former by finding a topological space $X$ such that the exponential $D^X$ exists, together with a surjective continuous map $X \to D^X$. Does there in fact exist such an $X$? 
Edit, 4/13/17: I'm still interested in this question, and so are some people associated with MIRI (at least when $n = 1$); for some details about why see here. 
 A: Here is a sort of partial solution. I doubt it will be very helpful, but anyone who wants to read it is free to do so.
Let $I$ be any space with the fixed point property. We will construct a space $X$ and a map $X \to I^X$ such that many interesting maps lie in its image, including all maps that can be defined in a language consisting of $e$, continuous functions from finite powers of $I$ to itself, and constant symbols in $X$ (which are all that is needed to prove the Lawvere fixed point theorem).
To accomplish this we will inductively construct for each natural number $n$ a space $X_n$, a map $e_n: X_n \times X_n \to I$, and a map $i_n: X_n \to X_{n+1}$, such that $e_{n+1}(i_n(x_1),i_n(x_2))=e_n(x_1,x_2)$. We will also use auxiliary spaces $Y_n,Z_n$ along the way.
After this we set $X$ to be the forward limit of $X_n$ along $i_n$ and let $e: X\times X \to I$ be the limit of the $e_n$.
The significance of the spaces $Y_n$ and $Z_n$ is that $Z_n$ is the set of continuous $I$-valued functions of $x \in X$ that depend only on $ e(x,x), e(x,t)$ for $t \in X_n$. $Y_n$ is the space of possible values of the pair $e(x,x), e(x,t)$, so that $Z_n$ is the space of functions on $Y_n$. Then $X_{n+1}$ is constructed so that it maps surjectively to $Z_n$, so all functions of $Z_n$ come from elements of $X_n$, and maps to $Y_n$, so that all functions of $Z_n$ can be extended to functions on $X_n$. For simplicity and canonicality, we define $X_{n+1}$ to be a subset of $Y_n \times Z_n$, defined by coherence condition to ensure the desired relationship between $Y_n$ and $e$.  Then we can freely add also the functions $e(t,x)$ for $t\in X_n$ to $Z_n$, as they are already continuous functions of $e(x,x)$, $e(x,t)$ by construction of $X_n$.
To begin, let $X_0$ be the empty set.
Inductively, assume we have defined $X_{n-1}, Y_{n-1},Z_{n-1}, X_n$.
Let $Y_n$ be $ I \times I^{X_n}$. Let $Z_n= I^{Y_n}$.
Let $X_{n+1} $ be the subset of $Y_n \times Z_n$ consisting of tuples $(b,c)\in Y_n, f\in Z_n$ satisfying the following two coherence conditions:


*

*$b= f(b,c)$

*For all $x$ in $X_n$, $c(x) = f(  e_n(x,x), t \mapsto e_n(x,t))$
Let $i_n: X_n \to X_{n+1}$ send $x= ((b',c'),f)$ to $(e_n(x,x), t \mapsto e_n(x,t) ) , (\beta,\gamma)\mapsto f'(\beta, \gamma \circ i_{n-1} )$.
Let $e_{n+1}: X_{n+1} \times X_{n+1} \to I$ send $((b_1,c_1),f_1),((b_2,c_2),f_2)$ to $f_1(b_2,c_2)$.

Let us first check that for $x$ in $X_n$ corresponding to a tuple $(b,c),f$ in $Y_{n-1} \times Z_{n-1}$, $i_n(x)$ satisfies the coherence conditions.
For $x$ in $X_n$ which corresponds to a tuple $(b,c),f$, we need $e_n(x,x) = f( e_n(x,x), t \mapsto e_n(x, i_{n-1}(t))$, but $e_n(x)=f(b,c)$ so it is sufficient to check that $b=e_n(x,x)$ and $c= t \mapsto e_n(x, i_{n-1}(t))$, which are the coherence conditions of $X_n$.
Furthermore we need for $x'$ in $X_n$ corresponding to a tuple $(b',c'),f)$, $e_n(x,x') = f( e_n(x',x'),t \mapsto e_n(x',i_{n-1}(t)))$, which is true because by definition of $e_n$, $e_n(x,x') = f(b',c')$ and $b' = f'(b',c') =e_n(x',x')$ while $c'(t)  =  f'( e_{n-1}(t,t), s \mapsto e_{n-1}(s,t)) = e_n(x', i_{n-1}(t))$ by definition of $e_n$ and $i_{n-1}$. 
Hence $i_n$ is actually a well-defined map from $X_n$ to $X_{n+1}$.

The fact that $e_{n+1}(i_n(x_1),i_n(x_2))=e_n(x_1,x_2)$ follows from unwinding the definitions.
Indeed, let $x_1 = ((b_1,c_1),f_1)$ and let $f_2=((b_2,c_2),f_2))$ and then  $e_{n+1} (i_n ( x_1),i_n(x_2)) = f_1 ( e_n (x_2,x_2),  t \mapsto e_n(x_2, i_{n-1}(t)) =f_1(b_2,c_2)= e_n(x_1,x_2)$ by the coherence conditions for $b_2,c_2$ and the definition of $e_n$, and then the definition of $e_n$.
Now we can define $X$ to be the forward limit of $X_n$ along $I_n$ and $e$ to be the forward limit of the maps $e_n$, which we have seen are compatible. Next we will characterize the image of the map $X \to I^X$ given by $x \mapsto (t \mapsto e(x,t))$. We will see that it this image can be viewed as the forward limit of $Z_n$ along the system of maps that we now define.

Consider the map $j_n: Y_{n+1} \to Y_{n}$ that sends $(b,c)$ to $( b, c \circ i_n)$. Consider also the map $k_n: Z_{n} \to Z_{n+1}$ by exponentiating $j_n$. 
I claim that if $i_{n} ((b,c),f)= ((b',c'),f')$ then $j_n(b',c') = (b,c)$ and $k_n(f)=f'$.
The first statement is simply the fact that $  b=e_n(x,x), c(t)= t \mapsto e_n(x,i_{n-1}(t))$, which follow from the definition of $e_n$ and the coherence conditions for $b$ and $c$ respectively.
The second statement follows immediately from the construction of $i_n$, which forces $f' ( \beta,\gamma) = f( \beta , \gamma \circ i_n) = \alpha(x)$ which is exactly $f \circ k_n$.
So we have verified the claim.  
Let $Y$ be the inverse limit of $Y_n$ along $j_n$ and let $Z$ be the inverse limit of $Z_n$ along $k_n$, so that there is a natural map $Y \times Z \to I$. The compatibilities of $i_n$ with $j_n$ and $k_n$ respectively imply that there are map $X \to Y$ and $X \to Z$, and $e: X \times X \to I$ is simply the composition of these two with the map $Y \times Z \to I$. 
Hence the map $X \to I^X$ induced by $e$ factors as $X \to Z \to I^Y \to I^X$. I can't prove that this composition is surjective but I can prove that the first map, $X \to Z$ is surjective.

To do this, it is sufficient to prove that the projection map $X_n \to Z_n$ is surjective. In other words, given $f: Y_n \to I$, construct $(b,c) \in Y_n$ satisfying the coherence conditions. Clearly we must have $c(x) = f(  e_n(x,x), t \mapsto e_n(x,t))$. Then we can take $b$ to be any fixed point of $f(b,c)$.

What maps can we construct this way? We get all the maps defined by elements of $Z_n$ for all $n$. These, in turn, are all continuous maps that depend only on the data in $Y_n$. By the coherence conditions and the definition of $e$, the projection of $x \in X_n$ to $Y_n$ encode $e(x,x),t\mapsto e(x,t)$ for $t$ in $X_{n-1}$. Using the definition of $e$ and the compatibility of $e$ with $i_{n-1}$, one can see that it also encodes $t \mapsto e(t,x)$ for $t \in Y_{n-1}$. The compatibility of $i_n$ with $e$ and with $j_n$ ensure that $Y_n$ continues to encode this data for all $x$. Hence all continuous functions that depend only on $t\mapsto e(t,x),e(x,x),t\mapsto e(x,t)$, which seem to me to be all the "obvious" ones that we construct, are in the image of the constructed map $X \to I^X$.
The definability statement from earlier follows from the fact that any finite set of elements of $X$, such as the constant symbols appearing in the formula, must lie in $X_n$ for some $n$. Any function that depends only on $e$ evaluated with these constant symbols will then lie in $Z_n$.
A: In my experience it is worth considering variants of Lawvere fixed point theorem. In the present case, I would split things up as follows, in order to circumvent the non-constructive nature of Brouwer's fixed point theorem.
Also, let me point out that we need not worry about exponentials too much, even though they do not exist in the category of topological spaces, unless the exponent is nice enough. We can move over to a cartesian-closed subcategory, such as teh compactly generated spaces, or to a cartesian closed supcategory, such as equilogical spaces.
Theorem: [Approximate Lawvere] Suppose $(B, d)$ is a metric space and $e : A \to B^A$ is a continuous map, such that for every continuous map $g : A \to B$ and $\epsilon > 0$ there is $a \in A$ such that $d(e(a)(a), g(a)) < \epsilon$. Then every continuous map $f : B \to B$ has approximate fixed points: for every $\epsilon > 0$ there is $b \in B$ such that $d(b, f(b)) < \epsilon$.
Proof. Given any $f$ and $\epsilon$ consider the map $g(a) = f(e(a)(a))$. there is $a \in A$ such that $d(e(a)(a), g(a)) < \epsilon$ and then $b = e(a)(a)$is an $\epsilon$-approximate fixed point of $f$. QED.
One way to use the theorem is via the sup metric (allowing infinite distance):
Corollary: If $e : A \to B^A$ has a dense image in the sup metric on $B^A$ then every endomap on $B$ has approximate fixed points.
Suppose we could apply the previous theorem to the closed ball $D^n$. Then we would know (constructively!) that every endomap on $D^n$ has approximate fixed points. then we just have another easy step, which contains all the classical reasoning needed:
Theorem: Suppose $X$ is compact and $f : X \to X$ has an $\epsilon$-approximate fixed point for every $\epsilon > 0$. Then $f$ has a fixed point.
Proof. By countable choice, for every $n$ there is $x_n \in X$ such that $d(x_n, f(x_n)) < 1/n$. Because $X$ is compact, $x_n$ has a subsequence converging to some $y \in X$. It is now easy to see that $y$ is a fixed point of $f$. QED.
But I do not see how to apply the approximate Lawvere to the closed ball, if that is even possible. 
A: EDIT: Pre-print available here. https://arxiv.org/abs/2005.01563 Note that I suspect that the use of the hypothesis of contractibility and the use of a homotopy is not really necessary, probably the hypothesis on the metric is enough.
Consider the "generalised Cantor space", the carrier set being $2^{\omega_{1}}$ and the basic open sets being sets consisting of all bit-strings of length $\omega_{1}$ with a fixed countable well-ordered bit-string as initial fragment. Denote this space by $A'$. Consider space with same carrier set with product topology obtained by viewing it as a product of $\aleph_1$ discrete spaces of cardinality two. Denote this space by $A''$. 
I define a space $X$ to be a ``nice space" if:
First, the space $X$ is compact and contractible and is the image of $A''$ under a continuous surjection. Secondly, the space $X$ is the disjoint union of an open dense subset $U$ and the complement $V$, and both $U$ and $V$ admit a ``homogenous" metric, where what we mean by this is as follows. Firstly, with regard to $U$, there exists some $\epsilon>0$, with the property that, for all $\delta$ such that $0<\delta<\epsilon$, every pair of distinct open balls of radius $\delta$ centred at a point in $U$ such that the closure of the balls does not intersect $V$, has the property that the balls in the pair are isometric. Then, with regard to $V$, we require that sufficiently small open balls in $X$, centred at points of $V$, of the same radius, are isometric.
Every closed ball in a finite-dimensional Euclidean space is indeed a nice space. The space $A''$ is a $k$-space, and so is every nice space, so exponential topologies exist here. I claim that for each nice space $X$ a continuous surjection $A' \rightarrow X^{A''}$ exists with the following property. Take an open subset of $X$ and take the pre-image under the evaluation map in $A'' \times X^{A''}$, then take projection onto the second factor and then pre-image of that in $A'$ under the continuous surjection $A' \rightarrow X^{A''}$. The result subset of $A'$ is open in $A'$, and its image under the continuous map $A' \rightarrow A''$ which fixes every point is open in $A''$. This is enough to apply the method of proof of Lawvere to obtain that every continuous endomorphism of $X$ has a fixed point. Thus Brouwer can be recovered as a corollary of a suitable generalisation of Lawvere.
The question of the existence of a space $A$ with a continuous surjection $A \rightarrow X^{A}$ for every nice space $X$, or for every space $X$ in some class which includes every closed ball in a finite-dimensional Euclidean space, still remains an open problem.
Brief outline of method of proof for showing that the continuous surjection exists. 
It is possible to construct an $\omega$-sequence $S:=\{C_{n}:n \in \omega\}$ of coverings of $X$ by finitely many open balls, each covering $C_{n} \in S$ being such that it can be partitioned into two collections of open balls with each collection having the property that all of the balls in it are pairwise isometric, and also such that the mesh of the covering $C_{n}$ tends towards zero as $n$ goes to infinity. Let $T$ be the collection of all centres of open balls occurring in some covering $C_{n}$. 
An element of $X^{A''}$ can be coded for, non-uniquely, by a mapping from a countable collection of countable bit-strings $B$, closed under taking initial fragments, and such that every element of $2^{\omega_1}$ has some element of $B$, maximal under the "initial fragment" relation, as an initial fragment, into $X$, with bit-strings of successor length being mapped to elements of $T$.  
One can further require that the trace of the mapping into $X$ along each branch of $B$ is ``generalised Cauchy"; it satisfies an obvious generalisation of the Cauchy criterion for each fragment of the branch of limit length, relative to the metric on $X$ which we have been holding fixed throughout, and with the speed of convergence having a uniform lower bound across all fragments of branches of $B$ of a given limit length.
Consider set of all mappings from such a set $B$ into $X$ satisfying the ``generalised Cauchy criterion" in question; this is in one-to-one correspondence with an appropriate set of countable well-ordered bit-strings $D$ under an appropriate coding scheme. Every element of $X^{A''}$ is coded for by at least one such mapping which in turn can be coded for by a countable well-ordered bit-string; further argument is required to show that every countable well-ordered bit-string does indeed code for an element of $X^{A''}$. To show that part of it, need to use a transfinite induction argument showing that from every mapping from an appropriate set $B$ into $X$ satisfying the appropriate constraints we can recover a continuous map $2^{\alpha} \rightarrow X$ for each countable ordinal $\alpha$ ($2^{\alpha}$ having the product topology), and that when $\alpha$ is sufficiently large, this map $2^{\alpha} \rightarrow X$ determines the map $2^{\beta} \rightarrow X$ (also continuous) for all $\beta$ such that $\alpha \leq \beta \leq \omega_1$. To get this transfinite induction to work we need to use both the "generalised Cauchy criterion" which is assumed to hold for our map $B \rightarrow X$ and also the given hypotheses on the space $X$, including the existence of an appropriate kind of metric on $X$.
Then, having done that, you need to show that the coding scheme for such maps $B \rightarrow X$, each such map being represented by a countable well-ordered bit-string from $D$, can indeed be constructed in such a way that it induces a surjection $A' \rightarrow X^{A''}$, continuous relative to generalised Cantor space topology on $A'$, and exponential topology on $X^{A''}$ arising from product topology on $A''$, which does indeed have all the properties I claimed.
That in very brief outline is the proof. Perhaps this post will still be thought inappropriate since I have not given all details. Or on the other hand maybe it can be accepted as a partial answer to the question.
A: This is not an answer, but I will try to explain why I think that it is unlikely for such a space $X$ to exist.
If we replace $D$ by say, a sphere, then (using Lawvere's fixed point theorem and the fact that spheres do have fixed-point-free self-maps) such a space $X$ does not exist for the sphere. Now, I really don't see how to possibly use the fact that the disc is a disc in constructing the space $X$. So I am tempted to believe that your $X$ does not exist.
A: Edit: Will Sawin has pointed out some difficulties with this answer. I'm going to leave this up for at least a while, in case anyone has any ideas about repairing it. Or perhaps this could be a cautionary tale?
My answer is that there is no space $X$ admitting a continuous surjection $X \to D^X$. 
Following one of Andrej's suggestions, let's extend the context from $\text{Top}$ to the quasitopos $\text{Choq}$ of Choquet (aka pseudotopological) spaces. This is a convenient setting because quasitoposes are cartesian closed (so we never have to wonder about the existence of certain exponentials, as we would for $\text{Top}$), and moreover this won't alter the problem of the OP because the full inclusion $i: \text{Top} \to \text{Choq}$ preserves cartesian products and any exponentials that happen to exist in $\text{Top}$. A further convenience is that $\text{Choq}$ is concrete and even topological (over $\text{Set}$), and we may say that a map of Choquet spaces is surjective if its underlying function is, so that the inclusion $i$ also preserves surjective maps. (By concreteness, every surjective map is an epimorphism, and the converse is true in $\text{Choq}$.) 
Suppose that there is a topological space $X$ such that $D^X$ exists in $\text{Top}$ and there is a surjective continuous map $\phi: X \to D^X$. In $\text{Choq}$ we get an induced map $D^\phi: D^{D^X} \to D^X$. The idea now is to construct a retraction to $D^\phi$ which might suggestively be denoted $\text{Ran}_\phi: D^X \to D^{D^X}$ (think "right Kan extension"), by exploiting the fact that $D = [0, 1]^n$ is an internal sup-lattice in $\text{Choq}$. Granting this possibility for the moment, and putting $Z = D^X$, we now have that $D^Z$ is a retract of $Z$ (to simplify notation, call the retraction $r: Z \to D^Z$ and the section $s: D^Z \to Z$), which opens the door to the argument given by Sam Eisenstat over at the related thread: Is there a topological space X homeomorphic to the space of continuous functions from X to [0, 1]?. In detail, for variables $z$ of type $Z$ and $g$ of type $D^D$, introduce the fixed-point combinator $Y: D^D \to D$ (which will live by the way in $\text{Top}$, since $D$ is an exponentiable space) by the formulas 
$$H := \lambda g. s(\lambda z. g(r(z)(z)))$$ 
$$Y := \lambda g. (r(H(g)))(H(g))$$ 
and verify in the usual manner that $Y(g)$ of type $D$ is a fixed point of $g$. Then invoke Sam's argument that such continuous fixed-point combinators $Y$ can't exist. (Technically, he indicated the reason for the special case $D = I = [0, 1]$, but the problem for general $D$ can be reduced to the special case as follows. Supposing a fixed point combinator $Y: D^D \to D$ exists, choose a pair $\rho: D \to I, \sigma: I \to D$ that exhibits $I$ as a retract of $D$, and verify that the composition 
$$I^I \stackrel{\sigma^\rho}{\to} D^D \stackrel{Y}{\to} D \stackrel{\rho}{\to} I$$ 
is a fixed point combinator, which is impossible by Sam's argument.) Contradiction. 
The task now is to construct the retraction $\text{Ran}_\phi: D^X \to D^{D^X}$. As indicated before, the background of this construction is a theory of sup-lattices in quasitoposes, for which I know no literature reference, but at some point I can make some notes on this available on my nLab web. In fact I had begun a pretty deep dive into this theory here, but then luckily found some simplifications which circumvent a lot of this theory. I'll go into the simplified proof in the final section, after introducing some necessary definitions in the next section (for those not very categorically minded, the next section has the function of sparing us the terrible agony of verifying that certain "easily defined" functions between certain Choquet spaces really are continuous maps -- the abstract account which comes next does all that work for us). 

A quasitopos $Q$ has a regular subobject classifier $\Omega$, which for $\text{Choq}$ is just a 2-point set $\{f, t\}$ equipped with the codiscrete (indiscrete) topology. (If $X$ is a Choquet space, then the points of $P X = \Omega^X$ are given by subspaces of $X$, i.e., subsets of $X$ with the subspace pseudotopology, or just the subspace topology if $X$ is a topological space.) Next, each map in $Q$ has a (unique) epi-(regular mono) factorization, due to the very useful fact that a quasitopos is not just regular but coregular ($Q^{op}$ is regular); see Johnstone's Elephant, Corollary A.2.6.3. The mono part will be called the regular image of $f$. This permits us to define, for a map $f: X \to Y$, the direct image mapping $\exists_f: \Omega^X \to \Omega^Y$ that in the case $Q = \text{Choq}$ takes a subspace $U$ of $X$ to the subspace $f(U)$ of $Y$. Formally, $\exists_f$ is defined by starting with the canonical regular subobject $\in_X \hookrightarrow X \times \Omega^X$ that is classified by the evaluation mapping $X \times \Omega^X \to \Omega$, then taking the regular image of the composite 
$$\in_X \hookrightarrow X \times \Omega^X \stackrel{f \times 1}{\to} Y \times \Omega^X,$$ 
then forming the classifying map $Y \times \Omega^X \to \Omega$ of the regular image, and then currying that to get $\exists_f: \Omega^X \to \Omega^Y$. 
Next, let us define a poset in $Q$ to be an object $X$ together with a regular subobject $i: X_1 \hookrightarrow X \times X$ satisfying standard axioms. If $X, Y$ are two posets, we can define their internal hom $[X, Y]$ as the pullback of an evident diagram 
$$\begin{array}{ccccc}
 & & & & Y_1^{X_1} \\ 
 & & & & \downarrow j^{X_1} \\ 
Y^X & \stackrel{sq}{\to} & (Y \times Y)^{X \times X} & \stackrel{(Y \times Y)^i}{\to} & (Y \times Y)^{X_1}
\end{array}
$$ 
where $sq(f) := f \times f$. The pullback of the regular mono $j^{X_1}$ along the bottom composite is again a regular mono $[X, Y] \to Y^X$. Of course $\Omega$ has a standard ordering $f \leq t$ and the points of $[X, \Omega]$ are given by regular upward-closed subobjects of $X$; for the quasitopos $\text{Choq}$ I'll just call them "up-sets", and similarly the points of $[X^{op}, \Omega]$ are "down-sets". 
We will want to consider $[X^{op}, \Omega]$ as a free posetal cocompletion of $X$, so let's say a few words on that. The "Yoneda embedding" $y_X: X \to [X^{op}, \Omega]$ is obtained by letting $\chi_{X_1}: X \times X \to \Omega$ classify the regular subobject $X_1 \hookrightarrow X \times X$, and then appropriately currying to a map $X \to \Omega^X$, and noting this factors through a map $X \to [X^{op}, \Omega]$ by the poset axioms. We form a cocompletion functor $\mathbf{P}$ which takes a poset $X$ to $[X^{op}, \Omega]$. For a poset map $f: X \to Y$, the map $\mathbf{P}f: [X^{op}, \Omega] \to [Y^{op}, \Omega]$ is to take a down-set $U$ of $X$ to the down-set generated by the direct image $\exists_f(U)$ in $Y$. Formally: if the poset $Y$ is given by a span $Y \stackrel{\pi_1}{\leftarrow} Y_1 \stackrel{\pi_2}{\to} Y$, then the composite 
$$\Omega^Y \stackrel{\pi_2^\ast}{\to} \Omega^{Y_1} \stackrel{\exists_{\pi_1}}{\to} \Omega^Y$$ 
has as its regular image the inclusion $[Y^{op}, \Omega] \hookrightarrow \Omega^Y$ we constructed earlier. This map maps a regular subobject to the down-set it generates. Denoting its epi-(regular mono) factorization as 
$$\Omega^Y \stackrel{e}{\to} [Y^{op}, \Omega] \hookrightarrow \Omega^Y$$ 
we now define $\mathbf{P}f: \mathbf{P}X \to \mathbf{Y}$ to be the composite 
$$[X^{op}, \Omega] \hookrightarrow \Omega^X \stackrel{\exists_f}{\to} \Omega^Y \stackrel{e}{\to} [Y^{op}, \Omega].$$ 
The functor $\mathbf{P}$ thus defined on the category of internal posets carries a monad structure whose unit is the Yoneda embedding $y$ (i.e., the component at $X$ is the principal down-set map $y_X: X \to [X^{op}, \Omega]$ we constructed earlier), and the multiplication turns out to be given by 
$$\text{mult}_X := [y_X^{op}, \Omega]: [[X^{op}, \Omega]^{op}, \Omega] \to [X^{op}, \Omega].$$ 
Officially, a sup-lattice is a $\mathbf{P}$-algebra. It can also be described as a poset $X$ whose Yoneda embedding $y_X: X \to \mathbf{P}X$ has a left adjoint (which will then be the algebra structure $\mathbf{P}X \to X$: there can be only one, as $\mathbf{P}$ is a lax idempotent or KZ monad). 

What seems to simplify matters greatly is to introduce another concept which, I'm pretty darned sure, is equivalent to the concept of sup-lattice. (See the post Retractions of Yoneda are reflectors, i.e., left adjoints? for some explanation why.) 
Definition: An s-lattice is a poset $X$ whose Yoneda embedding admits a retraction. We let $\sup_X$ generically denote a chosen retraction (although as I suggest, I think there can be at most one!). 
Here is the seed example. In $\text{Choq}$, the poset $I = [0, 1]$ is an s-lattice. The poset of down-sets $[I^{op}, \Omega]$ may be identified with $I \times \{f \leq t\}$ under the lexicographic order, topologized by the order topology. (One should check that the compact-open topology is in fact this order topology.) The Yoneda embedding takes $x \in I$ to $(x, t)$. The retraction is just the projection map $I \times \{f, t\} \to I$. 
Lemma 1: If $L$ is an s-lattice, then so is $L^X$ for any object $X$. 
Proof: Regard $X$ as an internal discrete poset (both maps of the span being $1_X$). There is an evident identification $\mathbf{P}(L \times X) = (\mathbf{P}L)^X$. Now examine the diagram 
$$\begin{array}{cccccc} 
L^X & \stackrel{y^X}{\to} & \mathbf{P}(L \times X) & & & & \\ 
y \downarrow & & y \downarrow & \searrow^{id} & & & \\ 
\mathbf{P}(L^X) & \underset{\mathbf{P}(y^X)}{\to} & \mathbf{PP}(L \times X) & \underset{\text{mult}}{\to} & \mathbf{P}(L \times X) & \underset{\sup_L^X}{\to} & L^X
\end{array}$$ 
(the square commutes by naturality of $y$, and the triangle commutes by a unit equation for the monad $\mathbf{P}$), and use the fact $\sup_L^X \circ y_L^X = 1_{L^X}$. Thus the bottom composite retracts $y_{L^X}$. $\Box$ 
For example, $D = I^n$ is an s-lattice in $\text{Choq}$. It follows further that any $D^X$ is an s-lattice. 
Recall that if $A, B$ are posets and $f: A \to B$ is a poset map, then $f$ is an embedding if $a \leq a'$ in $A$ whenever $f(a) \leq f(a')$ in $B$. It is easy to see that embeddings are monomorphisms. 
Lemma 2: If $f: X \to Y$ is a surjection in $\text{Choq}$ and $D$ is a poset, then $D^f: D^Y \to D^X$ is an embedding. 
This is pretty obvious. $D^f(g) := g \circ f$, so $D^f(g) \leq D^f(g')$ means $g f \leq g' f$. This implies $g \leq g'$ by surjectivity of $f$. 
Lemma 3: If $h: A \to B$ is a poset embedding in $\text{Choq}$, then $[h^{op}, \Omega]$ retracts $\mathbf{P}h$. 
Proof: It suffices to check this at the level of underlying sets, by concreteness of $\text{Choq}$ over $\text{Set}$. Let $U \in \mathbf{P}A$ be a down-set of $A$. Then $\mathbf{P}h(U) = \{b \in B: \exists_{a \in A} a \in U \wedge b \leq h(a)\}$. The function $[h^{op}, \Omega]$ sends this down-set to $\{a' \in A: \exists_{a \in A} a \in U \wedge h(a') \leq h(a)\}$. Since $h$ is an embedding, this is the same as $\{a' \in A: \exists_{a \in A} a \in U \wedge a' \leq a\}$, but this is just $U$ since $U$ is downward-closed. $\Box$ 
Our task is completed with the following result. 
Theorem: If $f: X \to Y$ is a continuous surjection in $\text{Choq}$ and $D$ is an s-lattice, then $\text{Ran}_f := \sup_{D^Y} \circ [(D^f)^{op},\Omega] \circ y_{D^X}$ retracts $D^f$. 
Proof: We have 
$$\begin{array}{ccc}
1_{D^Y} & = & \sup_{D^Y} \circ y_{D^Y} \\ 
 & = & \sup_{D^Y} \circ [(D^f)^{op}, \Omega] \circ \mathbf{P}(D^f) \circ y_{D^Y} \\ 
 & = & \sup_{D^Y} \circ [(D^f)^{op}, \Omega] \circ y_{D^X} \circ D^f
\end{array}$$ 
where the first equation uses Lemma 1, the second uses Lemmas 2 and 3, and the third uses naturality of $y$. $\Box$ 
