Kerodon

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Theorem 3.1.3.1. Let $f: X_{} \rightarrow S_{}$ be a Kan fibration of simplicial sets, and let $i: A_{} \hookrightarrow B_{}$ be any monomorphism of simplicial sets. Then the induced map

$\operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} )$

is a Kan fibration.

Proof. By virtue of Remark 3.1.2.7, it will suffice to show that if $i': A' \hookrightarrow B'$ is an anodyne morphism of simplicial sets, then every lifting problem of the form

$\xymatrix@C =100pt{ A'_{} \ar [d]^{i'} \ar [r] & \operatorname{Fun}( B_{}, X_{} ) \ar [d] \\ B'_{} \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) }$

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 Remark 3.1.2.7, since the left vertical map is anodyne (Proposition 3.1.2.9) and the right vertical map is a Kan fibration. $\square$