Size issues in the reduction of Colimits to a Coequalizer of Coproducts. This is in regards to a proof in Emily Riehl's Category Theory in Context (Available free here), on page 97. This is what I understand of the proof:
For a locally small category $C$, we wish to define a limit in $\text{Set}^C$ (Functors from $C$ to $\text{Set}$). To do so we define the limit on objects of $C$, and show that it assembles into a limit on $\text{Set}^{ob C}$, and then we invoke Proposition 3.3.9 on page 93, to show that the forgetful functor $\text{Set}^C \to \text{Set}^{obC}$ strictly creates limits, from which the result follows.
My issue is that the statement of Proposition 3.3.9 (whose proof is in part an exercise) supposes that $C$ is a small category (and my solution of that exercise does use this). Am I missing something here or is there some subtlety that makes this work out?
 A: I believe the point is to use the last paragraph of 3.3.9 which talks about extending an $\text{ob}\,\mathcal A$-indexed family of limits to a functor on $\mathcal A$. Assuming you already have this family of limits, then there aren't really any size issues with regards to this construction. For a given $f:A\to B$ in $\mathcal A$, we turn the limit cone $\lim_{J\in\mathsf J}D_A(J)$ over the diagram $D_A$ to a cone over $D_B$ which, by the universal property of $\lim_{J\in\mathsf J}D_B(J)$, induces an arrow $\lim_{J\in\mathsf J}D_A(J)\to\lim_{J\in\mathsf J}D_B(J)$ as desired. The only potentially sketchy part with respect to sizes in this is the mapping of cones induced by $f$. But while $\mathsf J\to\mathcal C^{\mathcal A}$ is sketchy given $\mathsf J$ is small and $\mathcal C$ and $\mathcal A$ are locally small, $\mathsf J\times\mathcal A\to\mathcal C$ and even $\mathcal A\to\mathcal C^{\mathsf J}$ are completely fine, and the latter is the functor we're using to transport (the bases of the) cones.
Other parts of 3.3.9 become questionable if $\mathcal A$ is not small, such as $\mathcal C^{\text{ob}\,\mathcal A}$ or $\prod_{\text{ob}\,\mathcal A}\mathcal C$ existing, at least as objects of $\mathbf{CAT}$. For the theorem you're discussing, we're only concerned about a particular functor/family, not the category of them. It's definitely a bit loose to talk about a functor $\mathbf{Set}^{\mathcal A}\to\mathbf{Set}^{\text{ob}\,\mathcal A}$, but the actual construction alluded to works functor-by-functor and that's all that's needed in this case.
