The Stacks project

Remark 77.13.2. In the situation of Lemma 77.13.1 denote

\[ F : \mathit{QCoh}(U, R, s, t, c) \to \mathit{QCoh}(\mathcal{O}_ U),\quad (\mathcal{F}, \beta ) \mapsto \mathcal{F} \]

the forgetful functor and denote

\[ G : \mathit{QCoh}(\mathcal{O}_ U) \to \mathit{QCoh}(U, R, s, t, c),\quad \mathcal{G} \mapsto (s_*t^*\mathcal{G}, \alpha ) \]

the right adjoint constructed in the lemma. Then the unit $\eta : \text{id} \to G \circ F$ of the adjunction evaluated on $(\mathcal{F}, \beta )$ is given by the map

\[ \mathcal{F} \to s_*s^*\mathcal{F} \xrightarrow {\beta ^{-1}} s_*t^*\mathcal{F} \]

We omit the verification.

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