# Kerodon

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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 4.4.3.18, and the equivalence $(3) \Leftrightarrow (5)$ follows from Theorem 4.4.4.4.