I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well.

A nice property of presentable $\infty$-categories is that if you have one such $\mathcal{C}$ and you have a full subcategory $\mathcal{D}\subseteq\mathcal{C}$ of it, then:

Theorem:The inclusion $\mathcal{D}\subseteq\mathcal{C}$ is reflective (i.e. has a left adjoint) if and only if $\mathcal{D}$ is closed under (small) limits and sufficiently filtered colimits.

This *almost* looks like the adjoint functor theorem, but note that we don't know that $\mathcal{D}$ is presentable in advance. In $1$-categories this is proved here, and I bet it is true $\infty$-categorically as well. Let's assume this for a moment.

Now, let $\mathbf{Pr}^L$ be the $\infty$-category of presentable $\infty$-categories and left adjoint (equivalently, colimit preserving) functors. I have a full subcategory $\mathcal{C}\subseteq \mathbf{Pr}^L$ and I want to know that there is a left adjoint to the inclusion, under some general closure conditions on $\mathcal{C}$. This would have been the case if only $\mathbf{Pr}^L$ was itself presentable...

Of course, there are some serious *size issues* here! The $\infty$-category $\mathbf{Pr}^L$ is *huge* (i.e. more than *large*, which is itself more than *small*) and I don't think it even has all *large* limits. What would be the poset $0 \to 1$ to the power of "a lot"? (at least its limit in the $\infty$-category of large categories is not presentable). Also, presentable $\infty$-categories are locally small, this doesn't seem to allow large limits. To conclude, I don't think $\mathbf{Pr}^L$ is "presentable" in the sense that we replace "small" with "large" and "large" with "huge". Keeping "small" and replacing only "large" with "huge" also doesn't seem to work for trivial reasons, since $\mathbf{Pr}^L$ can't be generated from a small set by small colimits, otherwise, it would have been at most *large*. Nevertheless, I still hope there might be a positive answer to the question in the title:

Question:Is there a characterization of reflective subcategories of $\mathbf{Pr}^L$ in terms of closure properties?

Perhaps this can be achieved by showing that $\mathbf{Pr}^L$ is "presentable" in some new sense, which takes into account the three sizes small-large-huge.

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