Kerodon

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Proposition 5.1.1.7. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $e$ is an isomorphism in $\operatorname{\mathcal{C}}$.

$(2)$

The morphism $e$ is $q$-cartesian and $q(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$.

$(3)$

The morphism $e$ is $q$-cocartesian and $q(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$.

Proof. We will prove the equivalence $(1) \Leftrightarrow (2)$; the proof of the equivalence $(1) \Leftrightarrow (3)$ is similar. The implication $(1) \Rightarrow (2)$ follows from Proposition 4.4.2.12 and Remark 1.4.1.6. To prove the converse, let $p: \operatorname{\mathcal{D}}\rightarrow \Delta ^0$ denote the projection map. If $q(e)$ is an isomorphism in $\operatorname{\mathcal{C}}$, then it is $p$-cartesian (Example 5.1.1.4). If, in addition, the morphism $e$ is $q$-cartesian, then it is also $(p \circ q)$-cartesian (Remark 5.1.1.6) and is therefore an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$ (Example 5.1.1.4). $\square$