What does it mean for a category to have ultra-powers? Paré, Makkai and myself seem to have independently come to the same conclusion that having ultra-powers is part of the axiomatics of a category of models of a theory written in first order logic. Preferably we should be able to state it as a diagonal functor having an adjoint as is the case with having limits and colimits. If X is a set and U is an ultrafilter on X, form a site X as follows: The underlying category is 2^X and the covers of I are finite families {J_i} so that . Then the characteristic function is a point of the site inducing an adjoint pair p_*, left adjoint to p^* : Shv( X ) -> Set. If we view p_*, as a diagonal; saying p^* exists is saying the U-ultrapower exists. If T is a coherent theory, p_* restricts to Mod(T) -> Mod(T, Shv( X )); but not in theories with negation (eg. the theory of sets with at least two elements). We seem to need Shv( X )/U for this. We loose adjoints in this construction; however perhaps this is the right thing interpreted over a different base topos other than sets, or viewed as a fibred category with a "non-standard" fibration.