Corollary 5.3.7.3 (0478)

Corollary 5.3.7.3. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets and let

\[ \operatorname{\mathcal{C}}_0 \xleftarrow { \pi } \operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \xrightarrow {\pi '} \operatorname{\mathcal{C}}_1 \]

denote the projection maps. Then:

$(1)$: If $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}$ are $\infty $-categories, then $\pi '$ is a cocartesian fibration of simplicial sets. Moreover, an edge $e$ of $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is $\pi '$-cocartesian if and only if $\pi (e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_0$.
$(2)$: If $\operatorname{\mathcal{C}}_1$ and $\operatorname{\mathcal{C}}$ are $\infty $-categories, then the evaluation map $\pi : \operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a cartesian fibration of simplicial sets. Moreover, an edge $e$ of $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is $\pi $-cartesian if and only if $\pi '(e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_1$.

Proof. Assertion $(1)$ follows by applying Proposition 5.3.7.2 in the special case $\operatorname{\mathcal{D}}_0 = \operatorname{\mathcal{D}}= \Delta ^0$ and $\operatorname{\mathcal{D}}_1 = \operatorname{\mathcal{C}}_1$. Assertion $(2)$ follows by a similar argument. $\square$