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The Stacks project

Lemma 7.10.14. The functor \textit{PSh}(\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{F} \mapsto \mathcal{F}^\# is exact.

Proof. Since it is a left adjoint it is right exact, see Categories, Lemma 4.24.6. On the other hand, by Lemmas 7.10.5 and Lemma 7.10.6 the colimits in the construction of \mathcal{F}^+ are really over the directed set \mathop{\mathrm{Ob}}\nolimits (\mathcal{J}_ U) where \mathcal{U} \geq \mathcal{U}' if and only if \mathcal{U} is a refinement of \mathcal{U}'. Hence by Categories, Lemma 4.19.2 we see that \mathcal{F} \to \mathcal{F}^+ commutes with finite limits (as a functor from presheaves to presheaves). Then we conclude using Lemma 7.10.1. \square


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