Warning 3.4.0.10. Let $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ be morphisms of simplicial sets, where $X$ is a Kan complex. The homotopy fiber products $X_0 \times ^{\mathrm{h}}_{X} X_1$ and $X_{1} \times _{X}^{\mathrm{h}} X_0$ are generally not isomorphic as simplicial sets. Instead, we have a canonical isomorphism
\[ (X_1 \times ^{\mathrm{h}}_{X} X_0)^{\operatorname{op}} \simeq X_{0}^{\operatorname{op}} \times ^{\mathrm{h}}_{ X^{\operatorname{op}} } X_1^{\operatorname{op}}. \]
However, $X_0 \times ^{\mathrm{h}}_{X} X_1$ and $X_{1} \times _{X}^{\mathrm{h}} X_0$ have the same weak homotopy type. To see this, we can use Proposition 3.1.7.1 to factor $f_0$ as a composition $X_0 \xrightarrow {w} X'_0 \xrightarrow {f'_0} X$, where $w$ is a weak homotopy equivalence and $f'_{0}$ is a Kan fibration. Using Propositions 3.4.0.7 and 3.4.0.9, we see that the maps
\[ X_0 \times ^{\mathrm{h}}_{X} X_1 \rightarrow X'_{0} \times _{X}^{\mathrm{h}} X_1 \hookleftarrow X'_0 \times _{X} X_1 \simeq X_1 \times _{X} X'_0 \hookrightarrow X_1 \times _{X}^{\mathrm{h}} X'_{0} \leftarrow X_1 \times _{X}^{\mathrm{h}} X_0 \]
are weak homotopy equivalences.