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Lemma 96.19.2. Generalities on relative Čech complexes.

1. If

   \[ \xymatrix{ \mathcal{V} \ar[d]_ g \ar[r]_ h & \mathcal{U} \ar[d]^ f \\ \mathcal{Y} \ar[r]^ e & \mathcal{X} } \]

   is $2$-commutative diagram of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$, then there is a morphism $e^{-1}\mathcal{K}^\bullet (f, \mathcal{F}) \to \mathcal{K}^\bullet (g, e^{-1}\mathcal{F})$.

2. if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism,

3. if $f, f' : \mathcal{U} \to \mathcal{X}$ are $2$-isomorphic, then the associated relative Čech complexes are isomorphic,