ScalingStacks

We first prove statement ([007B]) for \(\mathcal K= \emptyset\). Consider the locally coCartesian fibration \(\mathcal C^{\circledast} \rightarrow\mathbb V^{\circledast}\) from [Lur17, Not. 4.2.2.17, Lem.  4.2.2.20] associated to a \(\mathbb V\)-module category \(\mathcal C\). It follows from [Lur17, Lem.  4.8.4.12] that \(\mathrm{Fun}_{\mathbb V}(\mathcal C, \mathcal D) \subset \mathrm{Fun}_{/\mathbb V^{\circledast}}(\mathcal C^{\circledast}, \mathcal D^{\circledast})\) is the full subcategory on those functors which preserve locally coCartesian morphisms, where for given functors \(F\colon \mathcal A\rightarrow\mathcal B\leftarrow \mathcal C\colon G\) of \(\infty\)-categories, we let \(\mathrm{Fun}_{/ \mathcal B}(\mathcal A, \mathcal C) \coloneqq \mathrm{Fun}(\mathcal A, \mathcal C) \times_{\mathrm{Fun}(\mathcal A, \mathcal B)} \{F\}\) denote the over-functor category. If \(\mathcal C, \mathcal D\in \mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty})\) and \(\mathcal A\in \mathrm{Cat}_{\infty}\), the evident equivalence \[\mathrm{Fun}(\mathcal A, \mathrm{Fun}_{/\mathbb V^\circledast}(\mathcal C^{\circledast}, \mathcal D^{\circledast})) \simeq \mathrm{Fun}_{/\mathbb V^{\circledast}}(\mathcal A\times \mathcal C^{\circledast} , \mathcal D^{\circledast}) \simeq \mathrm{Fun}_{/\mathbb V^{\circledast}}((\mathcal A\times \mathcal C)^{\circledast} , \mathcal D^{\circledast})\] restricts to an equivalence \[ \mathrm{Fun}(\mathcal A, \mathrm{Fun}_{\mathbb V}(\mathcal C, \mathcal D)) \simeq \mathrm{Fun}_{\mathbb V}(\mathcal A\times \mathcal C, \mathcal D)\] which upon passing to maximal \(\infty\)-subgroupoids shows that \(\mathrm{Fun}_{\mathbb V}(\mathcal C, \mathcal D)\) is the morphism object for the action of \(\mathrm{Cat}_{\infty}\) on \(\mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty})\).