# Kerodon

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

Corollary 4.2.4.2. 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.