Lemma 6.154.
If \(C\) is compactly assembled and \(D\) is compactly generated, then \(C\otimes D\) is compactly assembled.
Proof
Choose a retraction of \(C\) from a compactly generated presentable category \(C'\). Tensoring with \(D\) gives a retraction of \(C\otimes D\) from \(C'\otimes D\). The tensor product of compactly generated presentable categories is compactly generated: if \(C'\simeq\Ind(C'_0)\) and \(D\simeq\Ind(D_0)\), it is generated under filtered colimits by the objects \(c\otimes d\) with \(c\in C'_0\) and \(d\in D_0\). The claim follows from Corollary 6.144.