$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 3.1.3.13. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & S, } \]
where $i$ is a monomorphism and $q$ is a Kan fibration. Then the simplicial set $\operatorname{Fun}_{A/ \, /S}(B, X)$ is a Kan complex. If $i$ is anodyne, then the Kan complex $\operatorname{Fun}_{A/ \, /S}(B,X)$ is contractible.
Proof.
By virtue of Remark 3.1.3.11, the simplicial set $\operatorname{Fun}_{A/ \, /S}( B, X)$ can be identified with a fiber of the restriction map
\[ \theta : \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,X). \]
Theorem 3.1.3.1 guarantees that $\theta $ is a Kan fibration, so its fibers are Kan complexes by virtue of Remark 3.1.1.9. If $i$ is anodyne, then $\theta $ is a trivial Kan fibration (Theorem 3.1.3.5), so its fibers are contractible Kan complexes (Remark 1.5.5.10).
$\square$