Re: Question on adjoints

[Steve Lack <s.lack@uws.edu.au>]

Dear Claudio,

Regarding

> 2) Anyway, the result states that in this case to say that "there is a natural
> isomorphism" is equivalent to say that "the canonical natural transformation
> (the counit) is an iso".
> Since many important categorical "exactness" conditions are expressed by
> requiring that some canonical transformations are iso (e.g. distributivity,
> Frobenius reciprocity and so on) one may wonder if also in these cases it is
> enough to require the existence of a natural isomorphism. I suppose that the
> answer is negative, but are there simple counter-examples?

my immediate response, like yours, was that this couldn't be true.

But in the case of distributivity, rather to my surprise, it turns out to be
true: any category with finite products and coproducts and a natural family
of isomorphisms XY+XZ~X(Y+Z) is in fact distributive.

The proof, unfortunately, will not fit in this margin, but I have typed some
notes which I can send you if you are interested.

Regards,

Steve Lack.
>
> Best regards,
>
> Claudio
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