Kerodon

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Proposition 8.3.3.2 (Existence and Uniqueness). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ admits a $\operatorname{Hom}$-functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ if and only if it is locally small. If this condition is satisfied, then $\mathscr {H}$ is uniquely determined up to isomorphism.

Proof. Combine Remark 8.3.5.3 with Corollary 5.6.0.6 (applied to the left fibration $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$). $\square$