Remark 8.2.0.5. Let $\operatorname{\mathcal{C}}_{-}$ be a (locally small) $\infty $-category. For every $\infty $-category $\operatorname{\mathcal{C}}_{+}$, Corollary 5.6.0.6 supplies an identification of equivalence classes of couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ (having essentially small fibers) with isomorphism classes of functors $T: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. Moreover, $\lambda $ is representable if and only if $T$ factors through the full subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ spanned by the representable functors (see Proposition 5.6.6.21). Consequently, Theorem 8.2.0.4 supplies a bijection
It is not hard to see that this bijection depends functorially on $\operatorname{\mathcal{C}}_{+}$, and is therefore induced by an isomorphism $\operatorname{\mathcal{C}}_{-} \simeq \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ in the homotopy category $\mathrm{h} \mathit{ \operatorname{QCat}}$. We can therefore regard Theorem 8.2.0.4 as an “implicit” version of Yoneda's lemma. We will give a more precise formulation in §8.3 (see Theorem 8.3.3.13).