An Equivalence Between Two Models of \(\infty \)-Categories of Enriched Presheaves

Abstract

Let \({{\mathcal {O}}}\rightarrow {\text {BM}}\) be a \({\text {BM}}\)-operad that exhibits an \(\infty \)-category \({{\mathcal {D}}}\) as weakly bitensored over non-symmetric \(\infty \)-operads \({{\mathcal {V}}}\rightarrow \text {Ass }, {{\mathcal {W}}}\rightarrow \text {Ass }\) and \({{\mathcal {C}}}\) a \({{\mathcal {V}}}\)-enriched \(\infty \)-precategory. We construct an equivalence

$$\begin{aligned} \text {Fun}_{\text {Hin}}^{{\mathcal {V}}}({{\mathcal {C}}},{{\mathcal {D}}}) \simeq \text {Fun}^{{\mathcal {V}}}({{\mathcal {C}}},{{\mathcal {D}}}) \end{aligned}$$

of \(\infty \)-categories weakly right tensored over \({{\mathcal {W}}}\) between Hinich’s construction of \({{\mathcal {V}}}\)-enriched functors of Hinich (Adv Math 367:107129, 2020) and our construction of \({{\mathcal {V}}}\)-enriched functors of Heine (Adv Math 417:108941, 2023).