Lemma 18.27.4. Internal hom and (co)limits. Let $\mathcal{C}$ be a category and let $\mathcal{O}$ be a presheaf of rings.
For any presheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor
\[ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \]commutes with arbitrary limits.
For any presheaf of $\mathcal{O}$-modules $\mathcal{G}$ the functor
\[ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O})^{opp} , \quad \mathcal{F} \longmapsto \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \]commutes with arbitrary colimits, in a formula
\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i, \mathcal{G}) = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}_ i, \mathcal{G}). \]
Suppose that $\mathcal{C}$ is a site, and $\mathcal{O}$ is a sheaf of rings.
For any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor
\[ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \]commutes with arbitrary limits.
For any sheaf of $\mathcal{O}$-modules $\mathcal{G}$ the functor
\[ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O})^{opp} , \quad \mathcal{F} \longmapsto \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \]commutes with arbitrary colimits, in a formula
\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i, \mathcal{G}) = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}_ i, \mathcal{G}). \]
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