The Stacks project

Lemma 90.16.6. 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 90.16.4 and the explanation of (S1) following Definition 90.10.1 it is immediate that (RS) implies (S1). This proves the first part of (S2). The second part of (S2) follows because Lemma 90.16.2 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 90.10.1. (In fact the morphism $y \to y'$ is compatible with both $a, a'$ and $c, c'$!) $\square$

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