Background By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is co …

Let $C,D$ be two presentable categories, $F,G : C \to D$ cocontinuous functors and $\eta :F \to G$ be a morphism of functors. Is it always true that the full subcategory $\{x \in …