The following is a very long comment and works in $1$-category theory.

> I claim that you can characterize very well coreflective subcategories. 

My strategy *works* even for reflective.

There is a biequivalence of $2$-categories
[$$ \text{Lex}^{op} \cong \text{Pres}. $$][1]

- Lex is the category of small categories with finite limits, $1$-cells are functors preserving finite limits.
- Pres is the category of finitely presentable categories, $1$-cells are accessible right adjoints.

Thus, $$\{\text{reflective subcategories of Pres}\} \leftrightarrow \{\text{coreflective subcategories of Lex}\}.$$


In a certain sense, the category Pres is the opposite of a locally presentable category. In fact, Lex is the category of algebras for the coKZ monad of free completion under finite limits (is this monad accessible?) on Cat. The category of algebras over an accessible monad defined on a locally presentable category is always locally presentable.

To conclude, as soon as one proves that Lex is locally presentable,  one can derive a lot of results for Pres, just because it's the opposite of an actual locally presentable category.



Achtung!

- Pres is not the same of $Pr^L$, nor its opposite category.
- The free completion under finite limits is not a monad, it is a $2$-monad and I am not expert enough in $2$-category theory to say anything about its algebras but *I strongly believe that Lex is finitely presentable as a $1$-category.*
- [Rosicky, Adameck and Trnkova][2] proved  that, **under Vopenka Principle, every subcategory closed under colimits in coreflective in a locally presentable category.** In fact this is equivalent to Vopenka.
[Rosicky][3] moved the theory of cotorsion theories to locally presentable categories.


  [1]: https://ncatlab.org/nlab/show/Gabriel-Ulmer+duality
  [2]: https://link.springer.com/article/10.1007%2FBF01182450
  [3]: https://ac.els-cdn.com/S0021869303006446/1-s2.0-S0021869303006446-main.pdf?_tid=1e050517-9105-4019-8282-fb4b806b47b6&acdnat=1532875524_b9319f2872e05dd867e975b56f300445