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I'd like a nice proof of the following fact.

Let $C$ and $D$ be categories, and let $\mathbf{Cat}/(C\times D)$ be the usual (1-categorical) slice category whose objects are triples $(X,F\colon X\to C, G\colon X\to D)$, $$C\xleftarrow{F}X\xrightarrow{G}D,$$ and whose morphisms are honest commuting triangles. That is, $$\mathrm{Hom}((X,F,G),(X',F',G'))=\{f\colon X\to X' \mid F'\circ f=F, G'\circ f=G\}.$$

Given $(X,F,G)$ as above, let $F^{\ast}\colon \mathbf{Set}^C\to \mathbf{Set}^X$ be the pullback functor and let $F_{\ast}$ be its right adjoint; similarly for $G$. Given a $C$-set $\delta\colon C\to \mathbf{Set}$, we will be interested in the continuous (polynomial-like) operation of moving it over to a $D$-set as $G_{\ast}F^{\ast}\delta$.

At this point we collect everything in sight into a functor, which might be called the evaluation functor $$K\colon (\mathbf{Cat}/(C\times D))^{op}\times \mathbf{Set}^C\to \mathbf{Set}^D$$ defined on objects as $$K((X,F,G),\delta):= G_{\ast}F^{\ast}\delta.$$

I want to show that $K$ preserves limits. I have convinced myself that this should work using a very low-level argument, but I am looking for efficient avenues by which to think about this, rather than small-minded plug and chug. A reference would be greatly appreciated. (Similarly, it seems true that the evaluation functor $L\colon (\mathbf{Cat}/(C\times D))\times \mathbf{Set}^C\to \mathbf{Set}^D\ $ given by $L((X,F,G),\delta)=G_!F^*\delta\ $ preserves colimits.)

I'd also be happy with an analogous theorem about polynomial functors in a locally cartesian closed category.

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Ok, this follows trivially from the construction of limits in ${\bf Cat}$.

Let $I\ $ be a small category, let $X\colon I\to{\bf Cat}\ $ denote a functor, and let $Y=\text{colim\}_{i\in I}X_i\ $ be its colimit with inclusion maps $q_i\colon X_i\to Y$. Let $d,e\colon Y\to {\bf Set}\ $ be any functors. The question posed here reduces to the question of whether the following function is an isomorphism: $$Hom_{Y-{\bf Set}}(d,e)\to \lim_{i\in I}\ Hom_{X_i-{\bf Set}}(q_i^{\ast}d,q_i^{\ast}e)$$

It indeed is an isomorphism because the category $Y-{\bf Set}\ $ of functors $Y\to{\bf Set}\ $ is the limit of the categories $X_i-{\bf Set}\ $, and we know that the objects and hom-sets of a limit in ${\bf Cat}$ are computed pointwise.

We now prove that the above observation implies that the functor $K$ above preserves limits. It suffices to fix each variable in the domain of $K$, because a limit in a product of categories is the product of the limits. Since $g_{\ast}$ and $f^{\ast}$ are right adjoints, $K$ is continuous in the second variable. It suffices to show that if $\delta\colon C\to{\bf Set}\ $ is any functor and $(Y,F,G)=\text{colim\}_{i\in I}(X_i,F_i,G_i)\ $ in ${\bf Cat}/(C\times D)$ then the map $$G_{\ast}F^{\ast}\delta\to\lim_{i\in I}(G_i)_{\ast}(F_i)^{\ast}\delta$$ is an isomorphism of $D$-sets.

Let $\epsilon\colon D\to{\bf Set}\ $ be a $D$-set. Using the Yoneda imbedding and the $(G^{\ast},G_{\ast})\ $ adjunction, it suffices to show that the function $$Hom_{Y-{\bf Set}}(F^{\ast}\delta,G^{\ast}\epsilon)\to\lim_{i\in I}Hom_{X_i-{\bf Set}}(F_i^{\ast}\delta,G_i^{\ast}\epsilon)$$ is a bijection, which was the observation above.

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The above "proof" is incomplete, and I am no longer convinced that the stated result is true. I don't know why it should suffice to "fix each variable in the domain of $K$". It was based on a wrong-headed idea, see Todd Trimble's comment in mathoverflow.net/questions/124361/… . I'm currently a bit embarrassed about this whole issue. –  David Spivak Mar 13 '13 at 2:57
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