Corollary 4.2.6.2. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. Then $f$ is a Kan fibration if and only if both of the evaluation maps
\[ \operatorname{ev}_{0}: \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 0\} , X) \times _{ \operatorname{Fun}( \{ 0\} , S)} \operatorname{Fun}( \Delta ^1, S) \]
\[ \operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 1\} , X) \times _{ \operatorname{Fun}( \{ 1\} , S)} \operatorname{Fun}( \Delta ^1, S) \]
are trivial Kan fibrations.