Proposition Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { A \ar [r]^{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^{g} \ar@ {-->}[ur]^{ \overline{f} } & S, } \]

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, 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 guarantees that $\theta $ is a Kan fibration, so its fibers are Kan complexes by virtue of Remark If $i$ is anodyne, then $\theta $ is a trivial Kan fibration (Theorem, so its fibers are contractible Kan complexes (Remark $\square$