The Stacks project

Lemma 18.14.3. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi.

  1. The functor $f_*$ is left exact. In fact it commutes with all limits.

  2. The functor $f^*$ is right exact. In fact it commutes with all colimits.

Proof. This is true because $(f^*, f_*)$ is an adjoint pair of functors, see Lemma 18.13.2. See Categories, Section 4.24. $\square$

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