ScalingStacks

Proof.

A projectively generated presentable \(\infty\)-category is also compactly generated since compact-projective objects are in particular compact. Thus, we only need to show that a left adjoint functor \(L\) between projectively generated presentable \(\infty\)-categories, which preserves compact-projective objects, also preserves compact objects. Indeed, by lemma 3.2.5.([003M]), \(L\) has a right adjoint which preserves sifted colimits and hence preserves filtered colimits. Applying the reverse direction of lemma 3.2.5.([003L]) now shows that \(L\) preserves compact objects. ◻