# Existence of product in the category of pre-sheaves of abelian categories

Let $X$ be $Top(X)$ be the category of open sets of $X$ with inclusion maps as morphism. Let $\mathcal{C}$ be abelian category and $\mathcal{C}_x$ denote the category of contravariant functors from $Top(X)$ to $\mathcal{C}$. Let $\mathcal{F}$ and $\mathcal{G} \in obj(\mathcal{C}_x)$. I want to show that product of $\mathcal{F}$ and $\mathcal{G}$ exist. Let us $\mathcal{F} \times \mathcal{G}(U)=\mathcal{F}(U) \times \mathcal{G}(U)$. I want to show this is the direct product. Let $\mathcal H \in \mathcal{C}_x$ with natural tranformations $i_1$ and $i_2$ to $\mathcal{F}$ and $\mathcal{G}$. Now, I know there is unique natural transformation $\eta$ from $\mathcal H$ to $\mathcal{F} \times \mathcal{G}$ as there is a unique map from $\mathcal H(U)$ to $\mathcal{F}(U) \times \mathcal{G}(U)$ for any open set $U$. But how do I show that naturality square for $\eta$ commutes?

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Commutes with the restriction maps? – Loki Clock Jun 21 '13 at 17:01
Yes with the restriction maps – Mohan Jun 21 '13 at 17:02

By the naturality square you mean the diagram you get with respect to some inclusion $V \to U$? If so then start with $x \in \mathcal H(U)$, push $x$, using $i_1$ and $i_2$ to $y \in \mathcal F(U)$ and $z \in \mathcal G(U)$. By naturality of $i_1$ and $i_2$ pushing $x|_V$ through $i_1$ and $i_2$ yields $y|_V$ and $z|_V$.
Now by definition of the maps $\mathcal H(U) \to \mathcal F(U) \times \mathcal G(U)$ we have that $x$ goes to $(y, z)$ and $x|_V$ goes to $(y|_V, z|_V)$ and by definition of the restrictions of $\mathcal{F \times G}$ we get that $(y, z)|_V = (y|_V, z|_V)$, thus $\mathcal{H \to F \times G}$ is natural.