Functorial Properties Preserved by Natural Isomorphism

Question

Conceptually, functors which are naturally isomorphic should have the same functorial properties e.g exactness, (co)continuity, etc. Thus, ideally, I'd hope for a precise definition of a functorial property and a meta theorem along the lines of

Naturally isomorphic functors have the same functorial properties.

Is there such a definition (and a metatheorem), and is what I said even true? Where can I find a precise formulation (or counterexamples)?

Answer 1

A precise definition of such a thing will be essentially tautological. Let me explain this on the (easier) example of groups suggested by Qiaochu Yuan in the comments. A precise (model theoretic) version of the statement

"Isomorphic groups have the same group-theoretic properties."

will be

If two models $\mathcal G,\mathcal G'$ of the theory of groups in the first-order language $\mathcal L = \{1,\times,{\cdot}^{-1}\}$ are isomorphic, then for any $\mathcal L$-sentence $\varphi$, $$\mathcal G \models \varphi \iff \mathcal G' \models \varphi .$$

And this is completely trivial, as an isomorphism between $\mathcal L$-structures $\mathcal G$ and $\mathcal G'$ is precisely a bijection $f \colon G \to G'$ such that for any formula $\varphi(x_1,\dots,x_n)$ and any tuple $(g_1,\dots,g_n) \in G^n$, $$ \mathcal G \models \varphi(g_1,\dots,g_n) \iff \mathcal G' \models \varphi(f(g_1),\dots,f(g_n)). $$

Here we have a precise statement, but we do not get anything from it.

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If you still want to work out the detail of your statement, all you need is to define a suitable (first-order) language and theory for functor.