Proof.
We will prove statement ([008J]), the proof of statement ([008K]) is completely analogous. Since \(\mathrm{RMod}_A(\mathbb V)\) is projectively generated by free modules \(v\otimes A\), see lemma 3.2.12.([004E]), where \(v \in \mathbb V\) is compact projective, it suffices to show that \(-\otimes_A M \colon \mathrm{RMod}_{A}(\mathbb V) \rightarrow\mathrm{RMod}_B(\mathbb V)\) preserves compact projective objects if and only if it sends such free modules \(v\otimes A\) to compact projectives in \(\mathrm{RMod}_B(\mathbb V)\) for all compacts \(v\in \mathbb V\). Since the action functor \(\mathbb V\times \mathrm{RMod}_B(\mathbb V) \rightarrow\mathrm{RMod}_B(\mathbb V)\) takes pairs of compact projectives to compact projectives by lemma 3.2.12.([004D]), this in turn is equivalent to the assertion that \(A\otimes_A M_B \simeq M_B \in \mathrm{RMod}_B(\mathbb V)\) is compact projective. ◻