My question is whether there are model-theoretic large cardinal axioms associated with $NFU$ - or more generally, with set theories other than $ZFC$.

Large cardinal properties generally come in one of three flavors - combinatorial (e.g. *Mahlo*), logical (e.g. *indescribability*), and model-theoretic (e.g. *measurable*). This last class has turned out to be in many ways the most fundamental - certainly, the model-theoretic large cardinal properties are the strongest (in terms of both consistency strength, straightforward logical strength, and - usually - size) by far. The prototypical model-theoretic large cardinal axiom is the following formulation of a measurable cardinal:

There is a nontrivial elementary embedding of $V$ into some inner model $M$.

The other model-theoretic large cardinal axioms assert the existence of elementary embeddings with additional nice properties - they "cohere" nicely, their codomains are large, etc.

I'm curious whether a similar phenomenon is known to exist with regard to some other set theory - let's take $NFU$ as the most likely example. It is not at all obvious to me that the existence of a nontrivial elementary embedding of $V$ into some inner model $M$ has any strength at all over $NFU$ (and in fact, it's not totally obvious to me what an "inner model" should even be!). So my question is: are there axioms which, when added to $NFU$, result in high consistency strength (say, at least that of $ZFC$ + a measurable), which are "model-theoretic" in flavor? Of course, this last requirement is very subjective; I'm hoping that, nonetheless, this question is meaningful.

I'm especially interested in model-theoretic large cardinal axioms over $NFU$ which seem not to have obvious $ZFC$ counterparts - that is, we can show that $NFU+(*)$ is consistent relative to large cardinals, and has consistency strength at least that of a measurable, but we can't pin down the exact strength, nor is there an obvious guess as to what it should be.

*To head off triviality, let's ignore axioms which are the relativization of a $ZFC$ large cardinal property to the Cantorian part of a model of $NFU$. :)*