Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 3.4.2.3 (Transitivity). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { A \ar [r] \ar [d] & B \ar [r] \ar [d] & C \ar [d] \\ A' \ar [r] & B' \ar [r] & C', } \]

where the left square is homotopy coCartesian. Then the right square is homotopy coCartesian if and only if the outer rectangle is homotopy coCartesian.

Proof. Apply Proposition 3.4.1.9. $\square$