# Reflections in locally presentable categories

In this paper, -4th line in the first paragraph on the (first) page 89,

then each full subcategory of $$\cal H$$ closed under limits ...

should or should not the word reflective be present:

then each full reflective subcategory of H closed under limits ... ?

I think that the word "reflective" is omitted by mistake, but I'm not sure.

It seems strange to me that if $$\cal H$$ is a locally presentable category then each full (but possibly non-reflective) subcategory of $$\cal H$$ closed under limits is closed under $$\alpha$$-filtered colimits for some regular cardinal $$\alpha$$.

• Have you tried looking at the reference mentioned in that sentence ? – Arnaud D. Jun 28 at 15:25
• @ArnaudD. I have managed to upload that reference number [9] here but I cannot locate in it that mentioned property. Could you please determine the relevant page for me? – user122424 Jun 28 at 15:42
• Is it not implied by Theorems 3 and 6 ? – Arnaud D. Jun 28 at 17:25

For every locally presentable category $$\mathcal{C}$$, every full subcategory $$\mathcal{D} \hookrightarrow \mathcal{C}$$ which is closed under limits is a reflective subcategory.