ScalingStacks

Composing with the symmetric monoidal left adjoint \({\mathbf K}^b\colon \mathrm{add}_{\mathbb{K}} \rightarrow\mathrm{st}_{\mathbb{K}}\) from proposition 4.2.7 induces a symmetric monoidal functor \[{\mathbf K}^b: \mathrm{add}_{\mathbb{K}}^J = \mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}}) \rightarrow\mathrm{Fun}(J^{\mathrm{op}}, \mathrm{st}_{\mathbb{K}}) =\mathrm{st}_{\mathbb{K}}^J\] left adjoint to the forgetful functor. Moreover, for \(\mathcal C\in \mathrm{add}_{\mathbb{K}}^J = \mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}})\), the unit of the adjunction \(\mathcal C\rightarrow{\mathbf K}^b(\mathcal C)\) is pointwise (i.e. for every \(j\in J\)) fully faithful.