Question on slice category

Question

Consider category $\mathcal{C}/Z$ consists of objects over Z, i.e arrows over $Z$. Let $h: H \rightarrow Z$ be a fixed object in $\mathcal{C}/Z$. Let $F=Hom(H,X)$. Show that under $F$, pullbacks over $Z$ are just products in the category of sets.

Answer 1

Here is a possible direction to solve the exercise:

1. prove that fiber products of morphisms with common codomain $Z$ are products in the category $\mathcal C/Z$
2. (Enter the Yoneda) use the fact that representable copresheaves preserve limit to conclude the proof.

If you need additional hints feel free to ask.

Addedum(since the OP asked :) ): Your functor $F$ is nothing but the representable functor associated to the object $h \colon H \to Z$, hence by general results it preserves product: that is it sends product diagrams in $\mathcal C/Z$ into product diagrams in $\mathbf{Set}$.

By (1) products in $\mathcal C/Z$ are fiber products in $\mathcal C$, hence putting all together we get that $F$ sends the fiber products of $\mathcal C$ (i.e. the products of $\mathcal C/Z$) into products of $\mathbf{Set}$.