$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 3.4.2.5. Suppose we are given a commutative diagram of simplicial sets
3.44
\begin{equation} \label{diagram:homotopy-pushout-square0} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r]^-{f'} & D } \end{gathered} \end{equation}
where $f$ is a weak homotopy equivalence. Then (3.44) is a homotopy pushout square if and only if $f'$ is a weak homotopy equivalence.
Proof.
For every Kan complex $X$, we obtain a commutative diagram of simplicial sets
3.45
\begin{equation} \label{diagram:pushout-lemma} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,X) & \operatorname{Fun}(B,X) \ar [l]_{u} \\ \operatorname{Fun}(C,X) \ar [u] & \operatorname{Fun}(D,X), \ar [l]_{u'} \ar [u] } \end{gathered} \end{equation}
where $u$ is a homotopy equivalence of Kan complexes (Corollary 3.1.7.5). Applying Corollary 3.4.1.3, we conclude that (3.45) is a homotopy pullback square if and only if $u$ is a homotopy equivalence of Kan complexes. Consequently, (3.44) is homotopy pushout square if and only if, for every Kan complex $X$, the composition with $f'$ induces a homotopy equivalence $\operatorname{Fun}(D, X) \rightarrow \operatorname{Fun}(C, X)$. By virtue of Corollary 3.1.7.5, this is equivalent to the requirement that $f'$ is a weak homotopy equivalence.
$\square$