Definition 18.16.1. With $u : \mathcal{C} \to \mathcal{D}$ satisfying (a), (b) above. For $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we define $g_{p!}\mathcal{F}$ as the presheaf

$V \longmapsto \mathop{\mathrm{colim}}\nolimits _{V \to u(U)} \mathcal{F}(U)$

with colimits over $(\mathcal{I}_ V^ u)^{opp}$ taken in $\textit{Ab}$. For $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we set $g_!\mathcal{F} = (g_{p!}\mathcal{F})^\#$.

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