Day and Lack's "Limits of small functors": Lemma 2.3

Question

I've been trying to understand the (4 line!) proof of Lemma 2.3 of Limits of small functors, on small functors into copresheaf categories $\mathbf{Set}^\mathcal C$. To me it seems to be using that the functor $\mathbf{Set}^\mathcal C \to \mathbf{Set}^{\text{ob } \mathcal C}$, defined by precomposing with the inclusion $\text{ob } \mathcal C \hookrightarrow \mathcal C$ with $\text{ob } \mathcal C$ the discrete category of objects, creates isomorphisms, which is false of course. So I'm wondering if I'm missing something.

Let $S\colon \mathcal K \to \mathcal M$ be a functor with $\mathcal M$ small cocomplete. $S$ is called small on p653 of 1 if there exists a small full subcategory $\mathcal B \subseteq \mathcal K$ and a natural isomorphism $S \cong \text{Lan}_J (S \circ J)$, where $J \colon \mathcal B \hookrightarrow \mathcal K$ is the inclusion and $\text{Lan}_J (S \circ J)$ is the left Kan extension of $S \circ J$ along $J$ (which exists by the assumption on $\mathcal M$).

Now if $\mathcal M = \mathbf{Set}^\mathcal C$ is a copresheaf category then there is a weaker notion as well: $S \colon \mathcal K \to \mathbf{Set}^\mathcal C$ is pointwise small if for each $C \in \mathcal C$ the composite $\text{ev}_C \circ S \colon \mathcal K \to \mathbf{Set}$ is small, where $\text{ev}_C \colon \mathbf{Set}^\mathcal C \to \mathbf{Set}$ is evaluation at $C$. The lemma asserts

Lemma. If $\mathcal C$ is small and $S \colon \mathcal K \to \mathbf{Set}^\mathcal C$ is pointwise small then $S$ is small.

Following the proof and using the pasting lemma for left Kan extensions (e.g. Theorem 4.47 of Kelly's book) it is straightforward to see that the hypotheses imply the existence of a small full subcategory $J \colon \mathcal B \hookrightarrow \mathcal K$ such that $\text{ev}_C \circ S \cong \text{Lan}_J (\text{ev}_C \circ S \circ J) \colon \mathcal K \to \mathbf{Set}$ simultaneously for all $C$. Now left Kan extensions into $\mathbf{Set}$ are defined pointwise, that is $\text{Lan}_J (\text{ev}_C \circ S \circ J)(Z)$ is a colimit for each $Z \in \mathcal K$ and, fixing $Z$, there is a unique way in which these colimits combine into a colimit that defines $\text{Lan}_J(S \circ J)(Z) \colon \mathcal C \to \mathbf{Set}$ (See sections X.3 and V.3 of "Categories work"). The uniqueness here is such that the family of universal cocones combines into the universal cocone defining the latter colimit and such that

$$\text{Lan}_J(S \circ J)(Z)(C) = \text{Lan}_J(\text{ev}_C \circ S \circ J)(Z).$$

We conclude that the left Kan extension $\text{Lan}_J(S \circ J)\colon \mathcal K \to \mathbf{Set}^\mathcal C$ satisfies

$$\text{Lan}_J(S \circ J)(Z)(C) \cong (\text{ev}_C \circ S)(Z) = S(Z)(C)$$

and we need these isomorphisms to combine into a natural isomorphism $S \cong \text{Lan}_J(S \circ J)$. But I don't see how they are natural in the variable $C$: there does not seem to be a reason for the unique functoriality of $\text{Lan}_J(S \circ J)(Z) \colon \mathcal C \to \mathbf{Set}$ to coincide with that of $S(Z)$. In particular there seems to be no reason for the family of natural isomorphisms $\text{ev}_C \circ S \cong \text{Lan}_{J_C} (\text{ev}_C \circ S \circ J_C) \colon \mathcal K \to \mathbf{Set}$, exhibiting the smallness of the $\text{ev}_C \circ S$, to be natural in $C$ in whatever sense.

Question 1. What am I missing here?

It can also be that I misunderstood the notion of smallness as follows. If the universal transformations defining the relevant left Kan extensions are canonical in the following way then the lemma does hold when the last two mentions of "small" are replaced by "nicely small":

Definition $S\colon \mathcal K \to \mathcal M$ is nicely small if there exists a small full subcategory $J\colon \mathcal B \to \mathcal K$ such that for all $Z \in \mathcal K$ the cowedge

$$ \mathcal K(Z, JB) \times (S \circ J)(B) \to SZ, $$

given by the action of $S$ on morphisms, defines $SZ$ as the coend $\int^{B \in \mathcal B} K(Z, JB) \times (S \circ J)(B)$.

It is straightforward to prove that nice smallness implies smallness. I've tried to prove the opposite but I seem to run into a "naturality issue" similar to the above one. With nice smallness instead of smallness one can prove the lemma: roughly, this is because the universal cowedges defining the $\text{ev}_C \circ S$ as left Kan extensions of their own restrictions are now given by the action of $S$ on morphisms in $Z$, which commutes with the action of $S$ on morphisms in $C$.

Question 2. I would like to hear any thoughts on the relation between nice smallness and smallness. Surely it is not the case that Day and Lack are implicitly using the notion of nice smallness instead of that of smallness?

Answer 1

Small and nicely small are indeed equivalent. I would consider this a fairly classical observation in the topic of small functor, at least when $M = Set$, so I wouldn't be surprised that Day and Lack are using it implicitly - maybe without realizing it - but I haven't had time to go re-read their paper in details to answer that part of the question. Of course in the case $M = Set$ the question of functionality in $c \in C$ doesn't appear, but my point is that the argument is that "nicely small objects are closed under small colimits" and is a purely "levelwise" thing when we had dependency in $c \in C$.

The idea is as follows:

I'm starting with the case of $M = Set$ first for simplicity, and also because this is the case I would consider a standard fact and on which I'm confident I won't say something false.

If $J$ is a small full subcategory of $K$, then we have an adjunction $L : Fun(J,Set) \leftrightarrows Fun(K,Set): R$ given by $L$ the Kan extension and $R$ the restriction. As Kan extensions are computed pointwise (and the coend version of the Yoneda Lemma I guess) the unit $X \to RL X$ is an isomorphism. So $L$ is a fully faithful inclusion. The condition of "nice smallness" you are expressing (with $J$ fixed) is expressing that the counit is an isomorphism, i.e. that the object $S$ is in the essential image of $L$.

This has two consequences:

1. If $S$ is nicely small for a given $J$, then it is also nicely small for any $J'$ such that $J \subset J'$. (because $L_J$ will factor through $L_{J'}$).

2. The full subcategory of nicely small objects with respect to $J$ is closed under small colimits (because $L$ is fully faithful and preserves colimits).

Put together this implies that nicely small objects are closed under any small colimits as given a small diagram of such one can always find a "small but big enough" $J$ that contains all the full subcategories of $K$ needed to see that all the objects appearing in the diagram are nicely small. It follows that any small colimit of representable is "nicely small", and hence any small functor is "nicely small".

Now, unless I'm completely missing the problem you are talking about, in the discussion above, everything seems to be nicely functorial if we add a parameter $c \in C$, and carries over immediately to the case of $M = Set^C$ hence solving your problem.