Editing Monad (category theory) (section)

=== Comonads ===
The [[Dual (category theory)|categorical dual]] definition is a formal definition of a ''comonad'' (or ''cotriple''); this can be said quickly in the terms that a comonad for a category <math>C</math> is a monad for the [[opposite category]]  <math>C^{\mathrm{op}}</math>. It is therefore a functor <math>U</math> from <math>C</math> to itself, with a set of axioms for ''counit'' and ''comultiplication'' that come from reversing the arrows everywhere in the definition just given.

Monads are to monoids as comonads are to ''[[comonoid]]s''. Every set is a comonoid in a unique way, so comonoids are less familiar in [[abstract algebra]] than monoids; however, comonoids in the category of vector spaces with its usual tensor product are important and widely studied under the name of [[coalgebra]]s.