$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 4.1.4.1. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, and let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. Then the restriction map
\[ \rho : \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S) \]
is also an inner fibration of simplicial sets.
Proof.
By virtue of Proposition 4.1.3.1, it will suffice 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, provided that $i'$ is inner anodyne. Equivalently, we must show that every lifting problem
\[ \xymatrix@C =100pt{ (A_{} \times B'_{} ) \coprod _{ A_{} \times A'_{} } ( B_{} \times A'_{} ) \ar [r] \ar [d] & X_{} \ar [d]^{q} \\ B_{} \times B'_{} \ar [r] \ar@ {-->}[ur] & S_{} } \]
admits a solution. This follows from Proposition 4.1.3.1, since the left vertical map is inner anodyne (Lemma 1.5.7.5) and $q$ is an inner fibration.
$\square$