Why does this setting imply that a category is Grothendieck?

Question

I came across the following Lemma in Mitsuyasu Hashimoto's Equivariant Twisted Inverses; it is Lemma 11.2 on page 107 of this pdf.

Let $\mathcal{A}$ be an abelian category which satisfies the (AB3) condition (i.e., is cocomplete) and $\mathcal{B}$ a Grothendieck category. Assume we have an adjoint pair $$ F \colon \mathcal{A} \rightleftarrows \mathcal{B} :G$$ such that $F$ is faithful exact and $G$ preserves filtered colimits. Then $\mathcal{A}$ is also Grothendieck.

Sadly, the proof is not given there but at least should work similar to the one before, where "Grothendieck" is replaced by "locally noetherian". Since one easily deduces the exactness of filtered colimits in $\mathcal{A}$, it remains to show that $\mathcal{A}$ admits a generator. But somehow I am confused between the notion of a generator/generating set in the sense of a separator (detecting non-trivial morphisms) and being able to write an object as a filtered colimit over a diagram involving only the generator. What is the connection between these notions?

Possible application: This statement would immediately give that the category of comodules over a flat Hopf algebroid $(A,\Gamma)$ is Grothendieck -- take $F$ to be the forgetful functor from $\Gamma$-comodules to $A$-modules and let $G$ be the cofree comodule.

Answer 1

Here's a rather convoluted proof of this lemma. It suffices to show that $\mathcal A$ is accessible, that is, $\mathcal A\simeq Ind_\kappa(\mathcal A_0)$ for some small category $\mathcal A_0$ (the sum of all objects in $\mathcal A_0$ is then a generator in $\mathcal A$). Since $F$ is faithful and exact, the adjunction is comonadic (Barr-Beck), and the comonad preserves filtered colimits by the assumption on $G$. In particular, the comonad is accessible, i.e., it preserves $\kappa$-filtered colimits for some regular cardinal $\kappa$. It is well-known that coalgebras for an accessible comonad on a locally presentable category form a locally presentable category: see Prop A.1 in http://www.math.harvard.edu/~eriehl/coalgebraic.pdf (this is basically a special case of the Makkai-Paré theorem stating that lax limits of accessible categories are accessible, Theorem 2.77 in Adámek-Rosicky).