I was talking about the following tentative argument. The 2-category of distributors (also called profunctors) $\mathrm{Dist}$ has (small) categories for objects. For $C,D:\mathrm{Dist}$ the 2-category of morphisms is defined as $$\mathrm{Dist}(C,D):=[D^{op} \times C; Set]$$
and denoted as $C \nrightarrow D$ (the middle line should be upright, but it's hard to latex). The order of arguments is chosen for the same variance as in the $Hom$-functor. The $Hom$-functor itself is the identity arrow $C \nrightarrow D$ The composition is given by the tensor product of bifunctors, i.e. if $F: D^{op} \times C \to Set$ and $G: E^{op} \times D \to Set$, then $$G\circ F(e, c) := \int^d G(e,d) \times F(d, c)$$ The fact that $Hom$ is identity is thus equivalent to the co-Yoneda lemma. There is an embedding $i: \mathrm{Cat} \to \mathrm{Dist}$, which is identity on objects and maps a functor $F: C \to D$ to the distributor $\hat F : C \nrightarrow D$ $$\hat F(d, c) = D(d, F c)$$
The distributors coming from the functors can be described as precisely the morphisms with a right adjoint, the right adjoint given by $\hat F^!(c, d) = D(F c, d)$. The inclusion $i: \mathrm{Cat} \to \mathrm{Dist}$ itself has a right adjoint which maps a category $C$ to the presheaf category $\hat C := [C^{op}, Set]$ and the distributors to their left Kan extension along the Yoneda embedding.

A more familiar 1-categorical construction is the allegory of correspondences $\mathrm{Corr}(S)$ in a 1-topos $S$. The only difference is that in allegories people usually say that morphisms are correspondences with a *left* adjoint. This is an unfortunate historical error due to the fact that correspondences are symmetric in both arguments, so the choice whether the graph of a function is a correspondence from $A$ to $B$ or vice versa is arbitrary. Since the subobjects of a set form a set, the adjunction realizes
$\mathrm{Corr}(S)$ as an algebraic category over $S$. For $S=Set$ this is the statement that the sets of subsets are precisely the boolean algebras. Unfortunately $\mathrm{Dist}$ doesn't seem to be algebraic or presentable over $\mathrm{Cat}$ in a naive way, since $\mathrm{Cat}$ is the category of *small* categories but the right adjoint maps $\mathrm{Dist}$ to *large* categories of presheaves. We don't even need an adjunction: presentable categories have small hom-sets while $[D^{op} \times C, Set]$ is a large category with a large core. Maybe some different notion of presentability is required?

The cartesian product $\times$ gives a symmetric monoidal structure on $\mathrm{Dist}$ and any category is dualizable (but not fully dualizable) w.r.t. this monoidal structure, the dual is the opposite category. This allows us to talk algebraically about both covariant and contravariant functors, as well as adjunctions. First let's talk about cartesian closed categories. We can encode the cartesian product as the morphism $C \times C \nrightarrow C$ which is a functor (=has right adjoint) and which is right adjoint to the diagonal $C\to C \times C$. The diagonal is a functor which arises by unversality of cartesian product w.r.t. functors. Similarly we can define a structure of $X^{\times n}$ for any $n:\mathbb{N}$. Thus we can encode a cartesian category as a tensor functor from the category which objects are finite sets with function to $\pm 1$ (negative will be required shortly) which is a tensor closed category, tensor product given by disjoint union of sets, a dual of $(\ast, +)$ is $(\ast, -)$ and vice versa, together with extra nontrivial generating morphisms which encode the functors and adjunctions in the definition of cartesian structure above. Now, any morphism $(\ast, +) \sqcup (\ast, +) \to (\ast, +)$ equivalently gives a morphism $(\ast, +) \to (\ast, -) \sqcup (\ast, +)$. The inner hom is defined by the condition that for the functor $C \times C \xrightarrow{\times} C$ this dual morphism is itself a functor, i.e. has a right adjoint. Adding the corresponding right adjoint to the classifying tensor category, we get a theory of cartesian closed categories.

Locally cartesian closed categories are defined similarly, but we need to work with arrow categories, so our tensor categories will include not only signed sets but also arbitrary signed finite categories. In particular, for lcc structure we focus on the arrow $\bullet \to \bullet$ and demand that for a fixed endpoint the overcategory has a cartesian closed structure (this will also involve diagrams of the form $\bullet \to \bullet \leftarrow \bullet$ etc). This shows that the lcc categories are equivalent to tensor functors from a certain tensor 2-category into $\mathrm{Dist}$. This would be sufficient for presentability if $\mathrm{Dist}$ was presentable, but as noted above it isn't quite so.

4more comments