Kerodon

Remark 5.2.1.4. In the situation of Proposition 5.2.1.3, the simplicial set $Z$ coincides with $\operatorname{Fun}( \Delta ^1,S) \times _{ \operatorname{Fun}(\{ 1\} , S) } \operatorname{Fun}(\{ 1\} , X)$ if and only if $q$ is a cartesian fibration. If this condition is satisfied, then Proposition 5.2.1.3 asserts that $\theta : Y \rightarrow \operatorname{Fun}( \Delta ^1, S) \times _{ \operatorname{Fun}( \{ 1\} , S) } \operatorname{Fun}( \{ 1\} , X)$ is a trivial Kan fibration.