Kerodon

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Proposition 4.5.3.5 (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 diagram. Then the right square is a categorical pushout diagram if and only if the outer rectangle is a categorical pushout diagram.

Proof. Apply Proposition 3.4.1.9. $\square$