Remark 11.9.4.4 (03PG)

Remark 11.9.4.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be locally small $\infty $-categories, with $\operatorname{Hom}$-functors

\[ \mathscr {H}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad \mathscr {H}_{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}. \]

Combining Corollary 11.9.4.3 and Proposition 8.3.4.1, we obtain homotopy equivalences

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})}( \operatorname{id}_{ \operatorname{\mathcal{C}}}, G \circ F) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})}( \mathscr {H}_{\operatorname{\mathcal{C}}}(-, -), \mathscr {H}_{\operatorname{\mathcal{C}}}( -, (G \circ F)(-) ) \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})}( \mathscr {H}_{\operatorname{\mathcal{D}}}(F(-), -), \mathscr {H}_{\operatorname{\mathcal{C}}}( -, G(-) ) \ar [u] .} \]

In particular, every natural transformation of functors $\eta : \operatorname{id}_{ \operatorname{\mathcal{C}}} \rightarrow G \circ F$ determines a natural transformation of profunctors $\eta ': \mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), -) \rightarrow \mathscr {H}_{\operatorname{\mathcal{C}}}( -, G(-) )$, which is characterized up to homotopy by the requirement that the diagram

\[ \xymatrix@R =50pt@C=50pt{ & \mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), F(-) ) \ar [dr]^-{ [\eta '] } & \\ \mathscr {H}_{\operatorname{\mathcal{C}}}(-,-) \ar [rr]^{ [\eta ] } \ar [ur]^{ [\gamma ] } & & \mathscr {H}_{\operatorname{\mathcal{C}}}( -, (G \circ F)(-) ) } \]

commutes in the homotopy category $\mathrm{h} \mathit{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }$.