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Definition 8.3.4.2. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small and let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(-, -): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. We say that a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is representable by $G$ if it isomorphic to the composition

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow {\operatorname{id}\times G} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{C}}}(-, -) } \operatorname{\mathcal{S}}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) ) \]

as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}^{<\kappa })$. By virtue of Proposition 8.3.3.2, this condition does not depend on the choice of $\operatorname{Hom}$-functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, - )$.