In Etale Cohomology Theory by Lei Fu, Lemma 8.5.7, the situation is the following: one has an etale map $j: U \to A$, and a compactifiable $g: A \to V$ and $g': U \to V$ the composite $g \circ j$. The morphisms $g$ and $g'$ are such that $Rg_!$ and $Rg'_!$ both make sense.
The author uses a "canonical" morphism $R^qg'_!\mathbb{Z}/n(d) \to R^qg_!\mathbb{Z}/n(d)$, where $\mathbb{Z}/n(d)$ is the constant sheaf on the appropriate space twisted by the roots of unity.
I am unsure of the definition of this morphism. My guess is that, since $Rg'_! \mathbb{Z}/n(d) \cong ((Rg_!) \circ j_!)\mathbb{Z}/n(d) $ and since $\mathbb{Z}/n(d) \cong j^* \mathbb{Z}/n(d)$, one has a morphism $j_!\mathbb{Z}n(d) \to \mathbb{Z}(d)$ by adjunction, and that the morphism is $R^qg'_!\mathbb{Z}/n(d) \to R^qg_!\mathbb{Z}/n(d)$ is the composite $R^qg'_!\mathbb{Z}/n(d) \to R^qg_!j_!\mathbb{Z}/n(d) \to R^qg_!\mathbb{Z}/n(d)$. Is this a correct definition of the morphism or does the author means another morphism?