The monoidal unit \(I \in J\) of any symmetric monoidal \(\infty\)-category \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) induces a symmetric monoidal functor \(\mathcal S\rightarrow\mathcal P(J)\) left adjoint to the evaluation functor \(\mathrm{ev}_{I} \colon \mathcal P(J) \rightarrow\mathcal S\), and explicitly given by sending a space \(X\) to the functor \(\mathrm{Hom}_{J}(-, I) \times X\colon J^{\mathrm{op}} \rightarrow\mathcal S\). It follows that for any presentably symmetric monoidal category \(\mathcal C\), there is a symmetric monoidal left adjoint \[\mathcal C\simeq \mathcal C\otimes \mathcal S\rightarrow\mathcal C\otimes\mathcal P(J) \simeq \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] to the evaluation functor \(\mathrm{ev}_{I}\colon\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C) \rightarrow\mathcal C\), explicitly given by sending \(c\in \mathcal C\) to the functor \(\mathrm{Hom}_{J}(-, I) \otimes c \colon J^{\mathrm{op}} \rightarrow\mathcal C\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2