Given \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) and \(A \in \mathrm{Alg}(\mathcal C)\), the following hold:
The \(\infty\)-category \(\mathrm{RMod}_A(\mathcal C)\) of right \(A\)-modules (see subsection A.9.1) is in \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\).
The action functor \(\mathcal C\otimes \mathrm{RMod}_A(\mathcal C) \rightarrow\mathrm{RMod}_A(\mathcal C)\) is in \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\), thus \(\mathrm{RMod}_A(\mathcal C) \in \mathrm{Mod}_{\mathcal C}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\).
If \(S\) is a set of compact projective generators of \(\mathcal C\), then the free modules \(\{c \otimes A_A\}_{c \in S}\) are compact projective generators of \(\mathrm{RMod}_A(\mathcal C)\).
Furthermore, if \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), then \(\mathrm{Mod}_A(\mathcal C)\) together with its symmetric monoidal relative tensor product is in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\).
Given \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) and \(A \in \mathrm{Alg}(\mathcal C)\), the following hold:
The \(\infty\)-category \(\mathrm{RMod}_A(\mathcal C)\) of right \(A\)-modules (see subsection A.9.1) is in \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\).
The action functor \(\mathcal C\otimes \mathrm{RMod}_A(\mathcal C) \rightarrow\mathrm{RMod}_A(\mathcal C)\) is in \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\), thus \(\mathrm{RMod}_A(\mathcal C) \in \mathrm{Mod}_{\mathcal C}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\).
If \(S\) is a set of compact generators of \(\mathcal C\), then the free modules \(\{c \otimes A_A\}_{c \in S}\) are compact generators for \(\mathrm{RMod}_A(\mathcal C)\).
Furthermore, if \(A \in \mathrm{CAlg}(\mathcal C)\), then \(\mathrm{Mod}_A(\mathcal C)\) together with its symmetric monoidal relative tensor product is in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\).