# Coreflective subcategories in Grothendieck/locally presentable categories

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 .