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?