On Wed, 24 Oct 2012 07:37:49 PM EDT "Vasili I. Galchin" <vigalchin@gmail.com> asked:

Lets consider presets P and P'. If we have adjoints between P and P' .. say (A, A'), what are the unit and co-unit natural transformations, respectfully?

If "presets" means pre-ordered sets (what I'd prefer to call just posets) then those natural transformations are just the order relations p </= a'a(p) (unit) and aa'(p') </= p' (co-unit) .

Likewise lets consider monoids M and M'. If we we have adjoints between M and M', (A, A') ... .what are the unit and co-unit natural transformation equations, respecfully?

If "respecfully" means "respectively", and if "monoid" means "category with one object", then the behavior of the functors A and A' is forced at the object level (each carries the only available object to the only available object), and the adjointness (M(*, A'*') = M'(A*, *')) forces M and M' to be isomorphic as monoids. On the other hand, if my guesses as to the intended meanings of those undefined terms are in error, then all bets are off :-) .

Regards,

Vasili

HTH. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]