Let \(\mathcal K\) be a small set of simplicial sets.
The \(\infty\)-category \(\mathrm{Cat}_{\infty}^{\mathcal K}\) is presentable and admits a presentably symmetric monoidal structure, which can be characterized as follows: If \(\mathcal C, \mathcal D\in \mathrm{Cat}_{\infty}^{\mathcal K}\), the tensor product \(\mathcal C\otimes \mathcal D\) is equipped with a functor \(\mathcal C\times \mathcal D\rightarrow\mathcal C\otimes \mathcal D\) which preserves \(\mathcal K\)-colimits separately in both variables and which induces for all \(\mathcal E\in \mathrm{Cat}_{\infty}^{\mathcal K}\) an equivalence \[\mathrm{Fun}^{\mathcal K}(\mathcal C\otimes \mathcal D, \mathcal E) \rightarrow\mathrm{\mathrm{Fun}^{\mathcal K\times \mathcal K}}(\mathcal C\times \mathcal D, \mathcal E),\] where \(\mathrm{Fun}^{\mathcal K}(\mathcal C\otimes \mathcal D, \mathcal E)\) denotes the full subcategory of \(\mathrm{Fun}(\mathcal C\otimes \mathcal D,\mathcal E)\) on those functors which preserve \(\mathcal K\)-colimits and where \(\mathrm{Fun}^{\mathcal K\times \mathcal K}(\mathcal C\times \mathcal D, \mathcal E)\) denotes the full subcategory of \(\mathrm{Fun}(\mathcal C\times \mathcal D, \mathcal E)\) on those functors which preserve \(\mathcal K\)-colimits separately in both variables.
Let \(\mathcal K'\) be a small set of simplicial sets with containing \(\mathcal K\). Then the subcategory inclusion \(\mathrm{Cat}_{\infty}^{\mathcal K'} \rightarrow\mathrm{Cat}_{\infty}^{\mathcal K}\) admits a symmetric monoidal left adjoint \[\mathcal P_{\mathcal K}^{\mathcal K'}\colon \mathrm{Cat}_{\infty}^{\mathcal K} \rightarrow\mathrm{Cat}_{\infty}^{\mathcal K'}\] whose unit \(\mathcal C\rightarrow\mathcal P_{\mathcal K}^{\mathcal K'}(\mathcal C)\) for \(\mathcal C\in \mathrm{Cat}_{\infty}^{\mathcal K'}\) is a fully faithful functor.