ScalingStacks

Example 4.1.4.

Let \(\mathcal K\) be a set of simplicial sets and recall from \(\mathrm{Cat}_{\infty}^{\mathcal K}\) the presentably symmetric monoidal \(\infty\)-category of \(\infty\)-categories with \(\mathcal K\)-colimits and \(\mathcal K\)-colimit preserving functors. For \(\mathcal C, \mathcal D\in \mathrm{Cat}_{\infty}^{\mathcal K}\), the full subcategory \(\mathrm{Fun}^{\mathcal K}(\mathcal C, \mathcal D)\) of \(\mathrm{Fun}(\mathcal C, \mathcal D)\) on the \(\mathcal K\)-colimit preserving functors is closed under \(\mathcal K\)-colimits [Lur17, Rem. 4.8.4.14] and hence is an object of \(\mathrm{Cat}_{\infty}^{\mathcal K}\). It follows directly from the characterization of the tensor product in \(\mathrm{Cat}_{\infty}^{\mathcal K}\), see proposition 3.1.11, that \(\mathrm{Fun}^{\mathcal K}(\mathcal C, \mathcal D) \in \mathrm{Cat}_{\infty}^{\mathcal K}\) is the morphism object between \(\mathcal C\) and \(\mathcal D\) in \(\mathrm{Cat}_{\infty}^{\mathcal K}\) (cf. proof of [Lur17, Lem. 4.8.4.2]).