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I was wondering, if it was possible to fully define the natural isomorphisms without the language of category theory, but only with that of set theory.

I am not interested in natural transformations in general, but only interested in those between the concrete ones, e.g., groups, rings, fields, vector spaces, algebras, etc.

Is it possible?

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  • $\begingroup$ Even for concrete categories, other than expanding definitions to make quantifiers explicit, I doubt there's a clear non-categorical description of natural transformations... At the very least, Eilenberg and MacLane were dealing with relatively tangible examples in algebraic topology c. 1942 when they decided to formalize nascent category theory. In my own experience, being interested in very tangible, concrete bits of mathematics, eventually it has become clear that at least a small part (a "pidgin"?) of categorical language is really much more economical and explanatory than "set theory". $\endgroup$ – paul garrett Apr 3 '16 at 22:41
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    $\begingroup$ What do you want "natural" to mean here? $\endgroup$ – user247327 Apr 3 '16 at 23:19
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You can certainly say "a finite dimensional vector space $V$ is naturally isomorphic to its double dual in that, for any linear map $f:V\to W$ the linear maps $V\to W \to W^{**},V\to V^{**}\to W^{**}$ coincide." But it seems clear there's no way to talk about natural isomorphisms in general without using a notion that's equivalent to that of a functor.

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