Kerodon

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Variant 7.1.3.2. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories. We say that $\widetilde{\operatorname{\mathcal{C}}}$ is corepresented by an object $X \in \operatorname{\mathcal{C}}$ if there exists an initial object $\widetilde{X} \in \widetilde{\operatorname{\mathcal{C}}}$ satisfying $U( \widetilde{X} ) = X$. In this case, we say that $\widetilde{X}$ exhibits $\widetilde{\operatorname{\mathcal{C}}}$ as a left fibration corepresented by $X$. We say that a left fibration $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ is corepresentable if it it is corepresented by some object of $\operatorname{\mathcal{C}}$: that is, if the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ has an initial object.