The Stacks project

Lemma 88.16.6. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. The condition (RS) for $\mathcal{F}$ implies both (S1) and (S2) for $\mathcal{F}$.

Proof. Using the reformulation of Lemma 88.16.4 and the explanation of (S1) following Definition 88.10.1 it is immediate that (RS) implies (S1). This proves the first part of (S2). The second part of (S2) follows because Lemma 88.16.2 tells us that $y = x_1 \times _{d, x_0, e} x_2 = y'$ if $y, y'$ are as in the second part of the definition of (S2) in Definition 88.10.1. (In fact the morphism $y \to y'$ is compatible with both $a, a'$ and $c, c'$!) $\square$


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