ScalingStacks

For \(J\in \mathrm{Cat}_{\infty}\) and \(\mathcal C\in \mathrm{Pr}^\mathrm{L}\), the functor \(\mathcal C\times J \rightarrow\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\), \[ (c,j)\mapsto c \otimes \mathrm{Hom}_{J}(-, j) \in \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] (where \(\otimes\) denotes the action of \(\mathcal S\) on \(\mathcal C\) inherited from the presentability of \(\mathcal C\)) induces an equivalence \[ \mathcal C\otimes \mathcal P(J) \simeq \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] in \(\mathrm{Pr}^\mathrm{L}\) (where \(\otimes\) denotes the tensor product of \(\mathrm{Pr}^\mathrm{L}\)).