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.