Lemma 21.36.4. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the corresponding morphism of topoi. Let $\mathcal{O}_\mathcal {D}$ be a sheaf of rings and let $\mathcal{I}$ be an injective $\mathcal{O}_\mathcal {D}$-module. If $g_!^{Sh} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ commutes with fibre products^{1}, then $g^{-1}\mathcal{I}$ is limp.

**Proof.**
We will use the criterion of Lemma 21.14.5. Condition (1) holds by Lemma 21.36.1. Let $K' \to K$ be a surjective map of sheaves of sets on $\mathcal{C}$. Since $g_!^{Sh}$ is a left adjoint, we see that $g_!^{Sh}K' \to g_!^{Sh}K$ is surjective. Observe that

by our assumption on $g_!^{Sh}$. Since $\mathcal{I}$ is injective module it is limp by Lemma 21.15.1 (applied to the identity). Hence we can use the converse of Lemma 21.14.5 to see that the complex

is exact as desired. $\square$

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