Remark 24.3.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. We have

Let $\mathcal{A}$ be a graded $\mathcal{O}_\mathcal {C}$-algebra. The multiplication maps of $\mathcal{A}$ induce multiplication maps $f_*\mathcal{A}^ n \times f_*\mathcal{A}^ m \to f_*\mathcal{A}^{n + m}$ and via $f^\sharp $ we may view these as $\mathcal{O}_\mathcal {D}$-bilinear maps. We will denote $f_*\mathcal{A}$ the graded $\mathcal{O}_\mathcal {D}$-algebra we so obtain.

Let $\mathcal{B}$ be a graded $\mathcal{O}_\mathcal {D}$-algebra. The multiplication maps of $\mathcal{B}$ induce multiplication maps $f^*\mathcal{B}^ n \times f^*\mathcal{B}^ m \to f^*\mathcal{B}^{n + m}$ and using $f^\sharp $ we may view these as $\mathcal{O}_\mathcal {C}$-bilinear maps. We will denote $f^*\mathcal{B}$ the graded $\mathcal{O}_\mathcal {C}$-algebra we so obtain.

The set of homomorphisms $f^*\mathcal{B} \to \mathcal{A}$ of graded $\mathcal{O}_\mathcal {C}$-algebras is in $1$-to-$1$ correspondence with the set of homomorphisms $\mathcal{B} \to f_*\mathcal{A}$ of graded $\mathcal{O}_\mathcal {C}$-algebras.

Part (3) follows immediately from the usual adjunction between $f^*$ and $f_*$ on sheaves of modules.

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