Proposition 11.3.0.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories which is corepresented by an object $X \in \operatorname{\mathcal{C}}$. Let $Y$ be another object of $\operatorname{\mathcal{C}}$. Then $Y$ corepresents the left fibration $U$ if and only if it is isomorphic to $X$.
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Proof. Since $U$ is corepresented by $X$, there exists an initial object $\widetilde{X} \in \operatorname{\mathcal{E}}$ satisfying $U( \widetilde{X} ) = X$. If $U$ is also corepresented by $Y$, then we can choose another initial object $\widetilde{Y} \in \operatorname{\mathcal{E}}$ satisfying $U( \widetilde{Y} ) = Y$. Applying Corollary 4.6.7.15, we deduce that there exists an isomorphism $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty $-category $\operatorname{\mathcal{E}}$. 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 left 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 $\widetilde{Y}$ is also an initial object of $\widetilde{\operatorname{\mathcal{C}}}$ (Corollary 4.6.7.15). It follows that the left fibration $U$ is corepresentable by the object $Y = U( \widetilde{Y} )$. $\square$