ScalingStacks

Proof.

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 \(\mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty})\).

Now let \(\mathcal K\) be general. Let \(\mathcal A\in \mathrm{Cat}_{\infty}^{\mathcal K}\) and \(\mathcal C, \mathcal D\in \mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\), and let \(\otimes\) denote the action of \(\mathrm{Cat}_{\infty}^{\mathcal K}\) on \(\mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\). By definition of the action, it induces an equivalence \[ \mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal A\otimes \mathcal C, \mathcal D) \simeq \mathrm{Fun}^{\mathcal K\times\mathcal K}_{\mathbb V}(\mathcal A\times \mathcal C, \mathcal D),\] where \(\mathrm{Fun}^{\mathcal K\times \mathcal K}_{\mathbb V}(\mathcal A\times \mathcal C, \mathcal D) \subset \mathrm{Fun}_{\mathbb V}(\mathcal A\times \mathcal C, \mathcal D)\) denotes the full subcategory of \(\mathbb V\)-linear functors whose underlying functor \(\mathcal A\times \mathcal C\rightarrow\mathcal D\) preserves \(\mathcal K\)-index colimits separately in each variable. On the other hand, by the description of \(\mathcal K\)-indexed colimits in \(\mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D)\) [Lur17, Lem. 4.8.4.13], the equivalence ([007E]) restricts to an equivalence of full subcategories \[ \mathrm{Fun}^{\mathcal K}(\mathcal A, \mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D)) \simeq \mathrm{Fun}^{\mathcal K, \mathcal K}_{\mathbb V}(\mathcal A\times \mathcal C, \mathcal D).\] Composing ([007F]) and ([007G]) exhibits \(\mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D)\) as the morphism object of \(\mathcal C, \mathcal D\) in \(\mathrm{Cat}_{\infty}^{\mathcal K}\). This proves part ([007B]). Part ([007C]) follows now with observation 4.1.2 applied to the (symmetric) monoidal left adjoint \(\mathrm{Cat}_{\infty}^{\mathcal K} \rightarrow\mathrm{Mod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\). ◻