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Is it possible to define a map between two categories which preserves all products and binary equalizers and yet is not a functor, ie it does not satisfy one or more axioms of a functor? Further, if you replace the functor with maps between categories that preserve all products and binary equalizers, what do the axioms of an adjoint become? That is, can we have continuous maps between categories and adjoints but no functors? I guess I am asking this: if we consider "a category theory" where the notion(or the job) of the functor is replaced by "maps which preserve products and binary equalizers", how much category theory do we loose? Do we simply loose stuff that was just excess baggage that came from the fact that we are presenting the theory of categories in SET?

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For your first question: Take a random finite category with no products or equalizers at all, and a random automorphism of the data that preserves none of the structure, and bingo. – Kevin Buzzard Jun 10 '11 at 6:47
Are you saying that, given there are no products, any random map (structure preserving or not) will trivially preserve all the products, since there are none? – Ben Sprott Jun 11 '11 at 14:47
I think that is what Kevin was aiming at. – Martin Brandenburg Jun 11 '11 at 16:47
up vote 6 down vote accepted

It does not make much sense to talk about non-functors $F$ which preserve products because there is no canonical morphism $F(x \times y) \to F(x) \times F(y)$. Even if we say that we choose one, we cannot formulate compatibility properties by varying $x,y$. So this will be just some random isomorphism, perhaps something built up by weird applications of Zorn's Lemma and many uncanonical choices. This won't be an interesting notion. This won't be of any use in category theory. In fact, it does not fit into the general idea of category theory, where each new data is required to be connected or even compatible with the given data.

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