Remark 5.3.1.16 (Detecting Isomorphisms). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of $\infty $-categories, and let $\alpha : F \rightarrow G$ be a morphism in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$. The following conditions are equivalent:
- $(1)$
The morphism $\alpha $ is an isomorphism in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$.
- $(2)$
The image of $\alpha $ is an isomorphism in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$.
- $(3)$
The image of $\alpha $ is an isomorphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$.
- $(4)$
For each object $X \in \operatorname{\mathcal{C}}$, the induced map $\alpha _{X}: F(X) \rightarrow G(X)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{X}$.
- $(5)$
For each object $X \in \operatorname{\mathcal{C}}$, the induced map $\alpha _{X}: F(X) \rightarrow G(X)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}$.
The implications $(1) \Leftrightarrow (2)$ is immediate, the equivalences $(2) \Leftrightarrow (3)$ and $(4) \Leftrightarrow (5)$ follow from Corollary 4.4.3.19, and the equivalence $(3) \Leftrightarrow (5)$ follows from Theorem 4.4.4.4.