Example 21.39.3. Let u : \mathcal{C}' \to \mathcal{C} be a functor. Endow \mathcal{C}' and \mathcal{C} with the chaotic topology as in Example 21.39.1. The functors u, \mathcal{C}' \to *, and \mathcal{C} \to * where * is the category with a single object and a single morphism are cocontinuous and continuous. Let g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \pi ' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (*), and \pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (*), be the corresponding morphisms of topoi. Let B be a ring. We endow * with the sheaf of rings B and \mathcal{C}', \mathcal{C} with the constant sheaf \underline{B}. In this way
is an example of Situation 21.38.3. Thus Lemma 21.38.5 applies to g so we do not need to distinguish between g_! on modules or abelian sheaves. In particular Remark 21.38.7 produces canonical maps
whenever we have \mathcal{F} in \textit{Ab}(\mathcal{C}), \mathcal{F}' in \textit{Ab}(\mathcal{C}'), and a map t : \mathcal{F}' \to g^{-1}\mathcal{F}. In terms of the computation of homology given in Example 21.39.2 we see that these maps come from a map of complexes
given by the rule
with obvious notation.
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