Lemma 17.3.3. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces.

The functor $f_* : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ Y)$ is left exact. In fact it commutes with all limits.

The functor $f^* : \textit{Mod}(\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X)$ is right exact. In fact it commutes with all colimits.

Pullback $f^{-1} : \textit{Ab}(Y) \to \textit{Ab}(X)$ on abelian sheaves is exact.

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