Remark 96.19.1. We can define the complex $\mathcal{K}^\bullet (f, \mathcal{F})$ also if $\mathcal{F}$ is a presheaf, only we cannot use the reference to Sites, Section 7.45 to define the pullback maps. To explain the pullback maps, suppose given a commutative diagram
\[ \xymatrix{ \mathcal{V} \ar[rd]_ g \ar[rr]_ h & & \mathcal{U} \ar[ld]^ f \\ & \mathcal{X} } \]
of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ and a presheaf $\mathcal{G}$ on $\mathcal{U}$ we can define the pullback map $f_*\mathcal{G} \to g_*h^{-1}\mathcal{G}$ as the composition
\[ f_*\mathcal{G} \longrightarrow f_*h_*h^{-1}\mathcal{G} = g_*h^{-1}\mathcal{G} \]
where the map comes from the adjunction map $\mathcal{G} \to h_*h^{-1}\mathcal{G}$. This works because in our situation the functors $h_*$ and $h^{-1}$ are adjoint in presheaves (and agree with their counter parts on sheaves). See Sections 96.3 and 96.4.
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