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.

Yes. Anyway it would be great if someone could explain this in an answer more directly (without the whole theory of chapter 2 in LPAC). – Martin Brandenburg Nov 29 '12 at 9:25