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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