One of the possible equivalent definitions of measurable cardinals defines them to be the critical point of an embedding of the universe $V$ into a transitive class $M$. Recently I learned of Trnková and Blass' result (see here) that the existence of a measurable cardinal is equivalent with the existence of an exact functor on the category of sets not naturally isomorphic to the identity.
I was wondering whether these two concepts - exact functors on sets, and elementary embeddings - are connected in a deeper way. Moreover, I was wondering if there are similar results connecting other large cardinals with preservation results for functors.