Lemma 89.16.6. 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 89.16.4 and the explanation of (S1) following Definition 89.10.1 it is immediate that (RS) implies (S1). This proves the first part of (S2). The second part of (S2) follows because Lemma 89.16.2 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 89.10.1. (In fact the morphism $y \to y'$ is compatible with both $a, a'$ and $c, c'$!) $\square$

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