The Stacks project

Lemma 90.16.2. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $ satisfying (RS). Given a commutative diagram in $\mathcal{F}$

\[ \vcenter { \xymatrix{ y \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A_1 \times _ A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A. } } \]

with $A_2 \to A$ surjective, then it is a fiber square.

Proof. Since $\mathcal{F}$ satisfies (RS), there exists a fiber product diagram

\[ \vcenter { \xymatrix{ x_1 \times _ x x_2 \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A_1 \times _ A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A. } } \]

The induced map $y \to x_1 \times _ x x_2$ lies over $\text{id} : A_1 \times _ A A_1 \to A_1 \times _ A A_1$, hence it is an isomorphism. $\square$

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