There is a question:
In any category $\mathcal{C}$, consider the category $\mathcal{C}_Z$ of objects over $Z$. Let $h:T\to Z$ be a fixed object in this category. Let $F$ be the functor such that $$F(X)=\text{Mor}_Z(T,X)$$where $X$ is an object over $Z$, and $\text{Mor}_Z$ denotes morphisms over $Z$. Show that $F$ transforms fiber products over $Z$ into products in the category of sets. (Actually, once you have understood the definitions, this is tautological.)
Author says it is tautological, and the solution I found said $\text{Mor}_Z(T,-)$(actually, $\text{Hom}_Z(T,-)$) preserves product so trivial(...). So I cannot verify my answer is no lack of logical jump or flaw(I think there could be flaw and there are lack, because I'm just a 3rd year student who major math, not friendly with category theory.)
So, can you verify my answer and if I wrong, can you show me flawless answer? Any help would be welcomed.
My answer
Fiber product is just an another name of product on $\mathcal C_Z$, so it is sufficient to show $F$ preserves (fiber)product.
For objects $f:X\to Z, g:Y\to Z$ let $\phi:F(X\times_ZY)\to F(X)\times_{F(Z)}F(Y)$ as $\phi(r)=(p_1\circ r,p_2\circ r)$, where $p_1,p_2$ is pull-back of $g$ by $f$ and pull-back of $f$ by $g$.
(Here comes a lack what I cannot justify) For every $(r_1,r_2)\in F(X)\times_{F(Z)}F(Y)$, $f\circ r_1=g\circ r_2$, so there exists unique $r\in F(X\times_ZY)$ which satisfies $\phi(r)=(r_1,r_2)$, so $\phi$ is bijection.
So we can see $F(X)\times_{F(Z)}F(Y)$ as $F(X\times_ZY)$, so $F$ preserves product.