By [BS24, Prop. 3.10], this construction agrees with the Day convolution structure on functor categories, as e.g. defined in [Lur17, Rem. 2.2.6.8], also see [BS24, Thm. 3.1]. Explicitly, the tensor product of functors \(F\colon J^{\mathrm{op}} \rightarrow\mathcal C\) and \(G \colon J^{\mathrm{op}} \rightarrow\mathcal C\) is given by the left Kan extension of the functor \(J^{\mathrm{op}} \times J^{\mathrm{op}} \xrightarrow{F\otimes G}\mathcal C\) along the tensor product \(J^{\mathrm{op}} \times J^{\mathrm{op}} \rightarrow J^{\mathrm{op}}\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2