Lemma 21.12.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $a : U' \to U$ be a monomorphism in $\mathcal{C}$. Then for any injective $\mathcal{O}$-module $\mathcal{I}$ the restriction mapping $\mathcal{I}(U) \to \mathcal{I}(U')$ is surjective.

**Proof.**
Let $j : \mathcal{C}/U \to \mathcal{C}$ and $j' : \mathcal{C}/U' \to \mathcal{C}$ be the localization morphisms (Modules on Sites, Section 18.19). Since $j_!$ is a left adjoint to restriction we see that for any sheaf $\mathcal{F}$ of $\mathcal{O}$-modules

Similarly, the sheaf $j'_!\mathcal{O}_{U'}$ represents the functor $\mathcal{F} \mapsto \mathcal{F}(U')$. Moreover below we describe a canonical map of $\mathcal{O}$-modules

which corresponds to the restriction mapping $\mathcal{F}(U) \to \mathcal{F}(U')$ via Yoneda's lemma (Categories, Lemma 4.3.5). It suffices to prove the displayed map of modules is injective, see Homology, Lemma 12.27.2.

To construct our map it suffices to construct a map between the presheaves which assign to an object $V$ of $\mathcal{C}$ the $\mathcal{O}(V)$-module

see Modules on Sites, Lemma 18.19.2. We take the map which maps the summand corresponding to $\varphi '$ to the summand corresponding to $\varphi = a \circ \varphi '$ by the identity map on $\mathcal{O}(V)$. As $a$ is a monomorphism, this map is injective. As sheafification is exact, the result follows. $\square$

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