Remark 21.37.7. Assumptions and notation as in Situation 21.37.3. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$, let $\mathcal{F}'$ be an abelian sheaf on $\mathcal{C}'$, and let $t : \mathcal{F}' \to g^{-1}\mathcal{F}$ be a map. Then we obtain a canonical map

$L\pi '_!(\mathcal{F}') \longrightarrow L\pi _!(\mathcal{F})$

by using the adjoint $g_!\mathcal{F}' \to \mathcal{F}$ of $t$, the map $Lg_!(\mathcal{F}') \to g_!\mathcal{F}'$, and the equality $L\pi '_! = L\pi _! \circ Lg_!$.

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