$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets, let $B_{}$ be an arbitrary simplicial set, and let $\rho : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} )$ be the morphism induced by composition with $f$. If $f$ is a left fibration, then $\rho $ is a left fibration. If $f$ is a right fibration, then $\rho $ is a right fibration.