# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition 4.5.4.6. A commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} }$

is a categorical pushout square if and only if the induced diagram of opposite simplicial sets

$\xymatrix@R =50pt@C=50pt{ A^{\operatorname{op}} \ar [r] \ar [d] & A_0^{\operatorname{op}} \ar [d] \\ A_1^{\operatorname{op}} \ar [r] & A_{01}^{\operatorname{op}} }$

is a categorical pushout square.

Proof. Apply Remark 3.4.1.7. $\square$