The Stacks project

Lemma 89.10.6. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. Let $\mathcal{F}_{x_0}$ be the category cofibred in groupoids over $\mathcal{C}_\Lambda $ constructed in Remark 89.6.4.

  1. If $\mathcal{F}$ satisfies (S1), then so does $\mathcal{F}_{x_0}$.

  2. If $\mathcal{F}$ satisfies (S2), then so does $\mathcal{F}_{x_0}$.

Proof. Any diagram as in Definition 89.10.1 in $\mathcal{F}_{x_0}$ gives rise to a diagram in $\mathcal{F}$ and the output of condition (S1) or (S2) for this diagram in $\mathcal{F}$ can be viewed as an output for $\mathcal{F}_{x_0}$ as well. $\square$

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