# Kerodon

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

Proposition 4.5.4.8 (Transitivity). Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [r] \ar [d] & C \ar [d] \\ A' \ar [r] & B' \ar [r] & C', }$

where the left square is a categorical pushout. Then the right square is a categorical pushout if and only if the outer rectangle is a categorical pushout.

Proof. Apply Proposition 3.4.1.11. $\square$