Definition 11.4.0.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We say that a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is corepresentable by $X$ if there exists an initial object $\widetilde{X} \in \operatorname{\mathcal{E}}$ satisfying $U( \widetilde{X} ) = X$. In this case, we say that the object $\widetilde{X}$ exhibits $U$ as a left fibration corepresented by $X$. We say that the left fibration $U$ is corepresentable if it is corepresentable by $X$ for some object $X \in \operatorname{\mathcal{C}}$.
We say that a right fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is representable by $X$ if there exists a final object $\widetilde{X} \in \operatorname{\mathcal{E}}$ satisfying $U( \widetilde{X} ) = X$. In this case, we say that the object $\widetilde{X}$ exhibits $U$ as a right fibration represented by $X$. We say that the fibration $U$ is representable if it is representable by $X$ for some object $X \in \operatorname{\mathcal{C}}$.