Question

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.

Answer 1

Since $\{x\in C : \eta(x) \textrm{ is an isomorphism}\}$ is cocomplete (indeed closed under colimits in $C$) it remains to show that it is accessible.

First observe that the arrow category $D^{\cdot\to\cdot}$ is again locally presentable. Let $D^{\cdot\to\cdot}_\cong$ be its full replete subcategory determined by all isomorphisms. This subcategory is also locally presentable and closed under colimits in $D^{\cdot\to\cdot}$.

Now consider the functor $H:C\longrightarrow D^{\cdot\to\cdot}$ given by $H(x) = \eta(x)$ on objects and $H(u:c\to c') = (Fu,Gu):\eta(c)\to\eta(c')$ on morphisms. Then $H$ is again a cocontinuous functor between locally presentable categories and $\{x\in C : \eta(x) \textrm{ is an isomorphism}\}$ is exactly the full preimage of $D^{\cdot\to\cdot}_\cong$ under $H$. By Remark 2.50 of LPAC it is therefore also accessible.