What happens if you try to define the stabilization of a category as an ultralimit instead of a limit

Something like: given a collection of objects {Aᵢ}ᵢ indexed by I and morphisms {fⱼ}ⱼ indexed by J and an ultrafilter U (on both J and I?) is there an "ultralimit" L with maps {πᵢ : L → Aᵢ}ᵢ such that fⱼπᵢ=πₖ is true on an element of U for any element of L? Or something?

Do you still get Łoś's thm?

Well, to be sure, it was a rhetorical question. Since obviously almost everyone wants to know how the weak encabulator cardinals encluster to the strong encabulation system in the ultralimit.

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assumptions, we prove that the ultralimit of a sequence of rescaled real Hilbert geometries is isometric to the Hilbert metric space of an open convex projective subset over a Robinson field. This result allows us to prove that ideal points of the [2/4 of https://arxiv.org/abs/2503.22671v1]