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
$\{x \in C : F(x) \xrightarrow{\eta(x)} G(x) \text{ is an isomorphism}\}$
of $C$ is presentable? In fact I only want to know if this category is complete (which happens to be the case in many examples; where of course the inclusion to $C$ doesn't have preserve limits). If necessary, you may assume that $F,G$ preserve finite limits.