# Is every limit-closed, accessibly-embedded full subcategory of a presentable $\infty$-category reflective?

Let $$C$$ be a presentable $$\infty$$-category and let $$D\subseteq C$$ be a full subcategory closed under limits and sufficiently-filtered colimits. If $$D$$ is known to be accessible, then by the adjoint functor theorem it follows that the inclusion $$D \subseteq C$$ has a left adjoint (and it then also follows that $$D$$ is presentable). Is it really necessary to check that $$D$$ is accessible, though?

In ordinary category theory, this shortcut is available: Theorem 2.48 of Adamek and Rosicky states that if $$C$$ is a locally presentable 1-category and $$D \subseteq C$$ is a full subcategory closed under limits and $$\kappa$$-filtered colimits for some $$\kappa$$, then $$D$$ is accessible and so the inclusion $$D \subseteq C$$ has a left adjoint (and hence $$D$$ is in fact locally presentable).

The proof that $$D$$ is accessible uses the Adamek and Rosicky's Corollary 2.36, which says that if $$C$$ is accessible and $$D \subseteq C$$ is a full subcategory closed under $$\kappa$$-filtered colimits and $$\kappa$$-pure subobjects for some $$\kappa$$, then $$D$$ is accessible. As far as I know, the theory of pure subobjects in the $$\infty$$-categorical setting has not appeared in the literature, but I believe that this theorem continues to hold $$\infty$$-categorically. What I'm not so sure of is the second part of the proof: Adamek and Rosicky's Remark 2.31 which shows that a $$\kappa$$-pure subobject $$c$$ of an object $$d$$ in a locally $$\kappa$$-presentable category $$C$$ is contained in the smallest full subcategory $$D \subseteq C$$ containing $$d$$ and closed under limits and $$\kappa$$-filtered colimits.

Or perhaps there is an alternate proof not using the theory of pure subobjects?