The Stacks project

Lemma 19.7.1. With notation as above. Suppose that $\mathcal{G}_1 \to \mathcal{G}_2$ is an injective map of abelian sheaves on $\mathcal{C}$. Let $\alpha $ be an ordinal and let $\mathcal{G}_1 \to J_\alpha (\mathcal{F})$ be a morphism of sheaves. There exists a morphism $\mathcal{G}_2 \to J_{\alpha + 1}(\mathcal{F})$ such that the following diagram commutes

\[ \xymatrix{ \mathcal{G}_1 \ar[d] \ar[r] & \mathcal{G}_2 \ar[d] \\ J_{\alpha }(\mathcal{F}) \ar[r] & J_{\alpha + 1}(\mathcal{F}) } \]

Proof. This is because the map $i\mathcal{G}_1 \to i\mathcal{G}_2$ is injective and hence $i\mathcal{G}_1 \to iJ_\alpha (\mathcal{F})$ extends to $i\mathcal{G}_2 \to J(iJ_\alpha (\mathcal{F}))$ which gives the desired map after applying the sheafification functor. $\square$

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