A higher Mackey functor description of algebras over an $N_{\mathrm{\infty }}$-operad

Suppose $G$ is a finite group. In this paper, we construct an equivalence between the $\mathrm{\infty }$-category of algebras over an $N_{\mathrm{\infty }}$-operad $\mathcal{O}$ associated to a $G$-indexing system $\mathcal{I}$ and the corresponding $\mathrm{\infty }$-category of higher incomplete $\mathcal{I}$-Mackey functors with value in spaces. We use the universal property of the incomplete $(2,1)$-category of spans of finite $G$-sets ${\mathcal{A}}_{\mathcal{I}}$ to construct a functor from ${\mathcal{A}}_{\mathcal{I}}$ to the $2$-category of $\mathcal{I}$-normed symmetric monoidal categories of Rubin. We then show that the left Kan extension of the composition of this functor with the core functor along the Yoneda embedding is an equivalence.