Proposition 4.4.1. ([Lur17, Thm. 4.8.5.15, Rem. 4.8.4.9]).
Let \(\mathbb V\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) and \(A, B \in \mathrm{Alg}(\mathbb V)\).
The \(\infty\)-category \(\mathrm{RMod}_A(\mathbb V)\) carries a left action by \(\mathbb V\), and can be viewed as an object in \(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L})\). This defines a symmetric monoidal functor \[\mathrm{RMod}_{-}(\mathbb V) \colon \mathrm{Alg}(\mathbb V) \rightarrow\mathrm{Mod}_\mathbb V(\mathrm{Pr}^\mathrm{L}).\]
Given an \(A\)–\(B\) bimodule \(_{A}M_{B} \in {}_{A}\mathrm{BMod}_{B}(\mathbb V)\), tensoring with \(M\) over \(A\) \[- \otimes_{A}M_{B} \colon \mathrm{RMod}_{A}(\mathbb V) \rightarrow\mathrm{RMod}_{B}(\mathbb V)\] defines a cocontinuous \(\mathbb V\)-linear functor, i.e. an object in \(\mathrm{Fun}^L_{\mathbb V}(\mathrm{RMod}_A(\mathbb V), \mathrm{RMod}_B(\mathbb V))\). These assemble into an equivalence: \[{}_{A}\mathrm{BMod}_{B}(\mathbb V) \xrightarrow{\simeq} \mathrm{Fun}^L_{\mathbb V}(\mathrm{RMod}_A(\mathbb V), \mathrm{RMod}_B(\mathbb V)).\] Furthermore, composition of functors corresponds to the relative tensor product of bimodules.