The Stacks project

Remark 18.27.3. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites. Let $\mathcal{F}, \mathcal{G}$ be sheaves of $\mathcal{O}_\mathcal {D}$-modules. There is a canonical map

\[ f^*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{F}, \mathcal{G}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {C}}(f^*\mathcal{F}, f^*\mathcal{G}) \]

Namely, this map is adjoint to the map

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{F}, \mathcal{G}) \longrightarrow f_*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {C}}(f^*\mathcal{F}, f^*\mathcal{G}) \]

defined as follows. Say $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. For sections over $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ we use the map

\begin{align*} \Gamma (V, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{F}, \mathcal{G})) & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ V}(\mathcal{F}|_ V, \mathcal{G}|_ V) \\ & \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{u(V)}}(f^*\mathcal{F}|_{u(V)}, \mathcal{G}|_{u(V)}) \\ & = \Gamma (u(V), \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {C}}(f^*\mathcal{F}, f^*\mathcal{G})) \\ & = \Gamma (V, f_*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {C}}(f^*\mathcal{F}, f^*\mathcal{G})) \end{align*}

where for the arrow we use pullback by the morphism $(\mathcal{C}/u(V), \mathcal{O}_{u(V)}) \to (\mathcal{D}/V, \mathcal{O}_ V)$ induced by $f$.

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