ScalingStacks

Observation 4.1.2.

Let \(F \colon \mathbb V\rightarrow\mathbb W\) be a monoidal functor from a monoidal \(\infty\)-category to a closed monoidal \(\infty\)-category \(\mathbb W\) which is left adjoint to a functor \(G\). Then, the induced \(\mathbb V\)-action on \(\mathbb W\) is closed with morphism object \(G \underline{\mathrm{Hom}}_{\mathbb W}(w, w') \in \mathbb V\) for \(w, w' \in \mathbb W\). If \(\mathbb V\) is also closed monoidal, then for \(v, v' \in \mathbb V\), the map of spaces \(\mathrm{Hom}_{\mathbb V}(v,v') \rightarrow\mathrm{Hom}_{\mathbb W}(Fv, Fv')\) lifts¹⁵ along \(\mathrm{Hom}_{\mathbb V}(I, -)\colon \mathbb V\rightarrow\mathcal S\) to a \(\mathbb V\)-morphism \[\underline{\mathrm{Hom}}_{\mathbb V}(v,v') \rightarrow G \underline{\mathrm{Hom}}_{\mathbb W}(Fv, Fv').\]