My understanding from model theory is that, given groups A and B, the statement $ A \cong B $ implies that any for any first order statement $ P$ in the language of groups, $ P(A) \iff P(B) $ .
Can an analogous statement be made for naturally isomorphic functors F and G? IE is it true that $ F \cong G \implies P(F) \iff P(G)$ for some collection of propositions about functors?
I’ve been told that isomorphisms are the “right” notion of equivalence for algebraic structures because they “preserve structure,” and I’m struggling greatly to see why natural isomorphisms are the “right” notion of equivalence of functors.
I’ve seen explainations like:
Set functions $ X \times C \rightarrow D $ can be identified with functions $ X \rightarrow D^C $ . Morphisms in a category $ X$ can be identified with functors $ 2 \rightarrow X$ , so morphisms in the functor category $ D^C$ should be identified with functors $ 2 \rightarrow D^C$ . But in analogy with set functions, $ 2\rightarrow D^C$ ‘equals’ $ 2 \times C \rightarrow D$ from which the definition of natural transformations can be derived.
The above explanation intuitively relates to a basic property of set functions and products, but doesn’t on the surface tell me about what specifically is true/preserved about isomorphic functors.
The definition of natural transformation is forced if we mandate that $ Cat$ be cartesian closed, but again I fail to see the relation to preservation of ‘structure.’
I’ve seen references to the ‘principle of equivalence’ which seems to support the idea that isomorphic objects should be be indistinguishable in some language.
Michael Makkai proposed the Principle of Isomorphism, “all grammatically correct properties of objects of a fixed category are to be invariant under isomorphism”
How does the definition of natural isomorphisms make this statement true in the category $ D^C$ with functors as objects and morphisms as natural transformations? What’s invariant? If natural transformations represent a “change in perspective” analogous to conjugation, what is preserved between the perspectives?
Can anyone give a formalization of the ideas behind the responses at https://math.stackexchange.com/questions/1077895/functorial-properties-preserved-by-natural-isomorphism?
I’m really stuck here. I feel like I’m missing something obvious.
Further reading: https://math.stackexchange.com/questions/1432782/how-different-can-equivalent-categories-be, https://math.stackexchange.com/questions/1685227/why-not-just-define-equivalence-relations-on-objects-and-morphisms-for-equivalen, and Can skeleta simplify category theory?.