$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 4.2.5.4. Let $f: X_{} \rightarrow S_{}$ and $i: A_{} \rightarrow B_{}$ be morphisms of simplicial sets, and let
\[ \rho : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]
be the induced map. If $f$ is a left fibration and $i$ is left anodyne, then $\rho $ is a trivial Kan fibration. If $f$ is a right fibration and $i$ is right anodyne, then $\rho $ is a trivial Kan fibration.
Proof.
We proceed as in the proof of Proposition 4.2.5.1. Assume that $f$ is a left fibration and that $i$ is left anodyne; we will show that $\rho $ is a trivial Kan fibration (the dual assertion for right fibrations follows by a similar argument). Fix a monomorphism of simplicial sets $i': A' \hookrightarrow B'$; we wish to show that every lifting problem
\[ \xymatrix@C =100pt{ A'_{} \ar [d]^{i'} \ar [r] & \operatorname{Fun}( B_{}, X_{} ) \ar [d]^{\rho } \\ B'_{} \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) } \]
admits a solution. Equivalently, we must show that every lifting problem
\[ \xymatrix@C =100pt{ (A_{} \times B'_{} ) \coprod _{ A_{} \times A'_{} } ( B_{} \times A'_{} ) \ar [r] \ar [d] & X_{} \ar [d]^{f} \\ B_{} \times B'_{} \ar [r] \ar@ {-->}[ur] & S_{} } \]
admits a solution. This follows from Proposition 4.2.4.5, since the left vertical map is left anodyne (Proposition 4.2.5.3) and the right vertical map is a left fibration.
$\square$