Kerodon

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Proposition 7.1.3.5. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories which is represented by an object $Y \in \operatorname{\mathcal{C}}$. Let $X$ be another object of $\operatorname{\mathcal{C}}$. Then $X$ represents the right fibration $U$ if and only if it is isomorphic to $Y$.

Proof. Since $U$ is represented by $Y$, there exists a final object $\widetilde{Y} \in \widetilde{\operatorname{\mathcal{C}}}$ satisfying $U( \widetilde{Y} ) = Y$. If $U$ is also represented by $X$, then we can choose another final object $\widetilde{X} \in \widetilde{\operatorname{\mathcal{C}}}$ satisfying $U( \widetilde{X} ) = X$. Applying Corollary 7.1.2.17, we deduce that there exists an isomorphism $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$. Then $e = U( \widetilde{e} )$ is an isomorphism from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$.

For the converse, suppose that there exists an isomorphism $e: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $U$ is a right fibration, we can lift $e$ to a morphism $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$. Applying Proposition 4.4.2.11, we see that $\widetilde{e}$ is also an isomorphism, so that $\widetilde{X}$ is also a final object of $\widetilde{\operatorname{\mathcal{C}}}$ (Corollary 7.1.2.17). It follows that $X = U( \widetilde{X} )$ represents the right fibration $U$. $\square$