Kerodon

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Exercise 4.1.5.12. Let $f: X \rightarrow S$ be an inner covering map of simplicial sets and let $i: A \hookrightarrow B$ be any monomorphism of simplicial sets. Show that the restriction map

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

is also an inner covering map. If $i$ is inner anodyne, show that $\theta$ is an isomorphism.