This question is a reference request for the following result or two results, which I believe are rather easy to prove.


Lemma. Let $\mathcal K$ be a locally presentable category and $\mathcal A\subset\mathcal K$ be a coreflective full subcategory. Assume that the coreflector $C\colon\mathcal K\to \mathcal A$ is an accessible functor (e.g., when viewed as a functor $\mathcal K\to\mathcal K$; this means that there exists a cardinal $\lambda$ such that $C$ preserves $\lambda$-directed colimits). Then


1. The category $\mathcal A$ is locally presentable.
2. If $\mathcal K$ is a Grothendieck abelian category and $\mathcal A$ is closed under kernels in $\mathcal K$, then $\mathcal A$ is a Grothendieck abelian category, too.


Is there any relevant reference? I was only able to find Corollary 6.29 in the book of Adámek and Rosický "Locally presentable and accessible categories". This corollary claims, among other things, that any coreflective full subcategory $\mathcal A$ in a locally presentable category $\mathcal K$ is locally presentable, if one assumes Vopěnka's principle.


My lemma above does not depend on Vopěnka's principle or any other set-theoretical assumptions. Part 1. of it is an elementary version of this corollary from the book of Adámek and Rosický. Is there any other/better reference?


Some context: part 2. of the lemma is a generalization of Lemma 3.4 from my preprint S.Bazzoni, L.Positselski "Matlis category equivalences for a ring epimorphism", <https://arxiv.org/abs/1907.04973> .



Coreflective subcategories in Grothendieck/locally presentable categories
Since a coreflective full subcategory is the category of coalgebras for the induced idempotent comonad, 1. is answered in [presentability rank of categories of coalgebras](https://mathoverflow.net/questions/350351/presentability-rank-of-categories-of-coalgebras) (the corresponding comonad is accessible).


