Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 4.1.4.6. Suppose we are given a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & S } \]

of simplicial sets, where $i$ is a monomorphism and $q$ is an inner fibration. Then the simplicial set $\operatorname{Fun}_{A/ \, /S}(B, X)$ of Construction 3.1.3.7 is an $\infty $-category. Moreover, if $i$ is inner anodyne, then $\operatorname{Fun}_{A/ \, /S}(B, X)$ is a contractible Kan complex.

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

\[ \theta : \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,X). \]

Proposition 4.1.4.1 asserts that $\theta $ is an inner fibration of simplicial sets, so its fibers are $\infty $-categories (Remark 4.1.1.6). If $i$ is inner anodyne, then Proposition 4.1.4.4 guarantees that $\theta $ is a trivial Kan fibration, so its fibers are contractible Kan complexes. $\square$