homomorphisms of (pre)sheaves

DEFINITION 06.7   Let $\mathcal F_1$ , $\mathcal F_2$ be presheaves of modules on a topological space $X$ . Then we say that a sheaf homomorphism

$$\varphi:\mathcal F_1 \to \mathcal F_2$$

is given if we are given a module homomorphism

$$\varphi_U: \mathcal F_1(U) \to \mathcal F_2(U)$$

for each open set $U\subset X$ with the following property hold.

1. For any open subsets $V,U \subset X$ such that $V\subset U$ , we have
   $$\rho_{V,U} \circ \varphi_U=\varphi_V\circ \rho_{V,U}.$$

(The property is also commonly referred to as ``$\varphi$ commutes with restrictions''.)

DEFINITION 06.8   A homomorphism of sheaves is defined as a homomorphism of presheaves.