Corollary 11.9.4.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between locally small $\infty $-categories. Then, for any functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, precomposition with the natural transformation $\gamma $ of Proposition 11.9.4.1 induces a homotopy equivalence
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) }( \mathscr {H}_{\operatorname{\mathcal{D}}}(F(-), -), \mathscr {H}_{\operatorname{\mathcal{C}}}(-, G(-)) ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {H}_{\operatorname{\mathcal{C}}}(-,-), \mathscr {H}_{\operatorname{\mathcal{C}}}( -, (G \circ F)(-)) ). } \]