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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 $\pi$ 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. $\square$