$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 3.4.0.9. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [r] \ar [d] & X \ar [d] & X_1 \ar [l] \ar [d] \\ Y_0 \ar [r] & Y & Y_1 \ar [l] } \]
where $X$ and $Y$ are Kan complexes and the vertical maps are weak homotopy equivalences. Then the induced map $X_0 \times _{X}^{\mathrm{h}} X_1 \rightarrow Y_0 \times _{Y}^{\mathrm{h}} Y_1$ is also a weak homotopy equivalence.
Proof.
Apply Proposition 3.4.0.2 to the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\Delta ^1, X) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, X) \ar [d] & X_0 \times X_1 \ar [l] \ar [d] \\ \operatorname{Fun}( \Delta ^1, Y) \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, Y) & Y_0 \times Y_1, \ar [l] } \]
noting that the left horizontal maps are Kan fibrations by virtue of Corollary 3.1.3.3.
$\square$