Example 3

Let B be a topological space, $\mathscr{B}$ a basis of the topology of B and I a set. For every $i\in I$, let $\mathscr{F}_i= (\mathscr{F}_i(U), f_{i,UV})$ be a presheaf on B relative to $\mathscr{B}$. Denote by $\mathscr{F}$ the product presheaf of the family $(\mathscr{F}_i)_{i\in I}$. For every open set $U\in \mathscr{B}$, one has $\mathscr{F}(U) =\prod_{i\in I}\mathscr{F}_i(U)$; for every $i\in I$, denote by pr$_{i,U}:\mathscr{F}(U)\rightarrow \mathscr{F}_i(U)$ the projection of index $i$. It follows immediately from the definition of the presheaf $\mathscr{F}$ that the family pr$_i$ = (pr$_{i,U}$)$_{U\in\mathscr{B}}$ is a morphism of presheaves from $\mathscr{F}$ into $\mathscr{F}_i$. The morphism pr$_i$ is called the projection morphism of index $i$. For every presheaf $\mathscr{F}'$ on B relative to $\mathscr{B}$ and every family $(\psi_i)_{i\in I}$, where $\psi_i$ is a morphism of presheaves from $\mathscr{F}'$ into $\mathscr{F}_i$, there exists a unique morphism of presheaves $\psi :\mathscr{F}'\rightarrow \mathscr{F}$ such that for every $i\in I$, pr$_i\circ \psi =\psi_i$.