  Almost any reasonable class of finite relational structures has the Ramsey
property or a Ramsey expansion. In contrast to that, the list of classes of
finite algebras with the Ramsey expansion is surprisingly short. In this paper
we show that any nontrivial variety (that is, equationally defined class of
algebras) enjoys various \emph{dual} Ramsey properties. We develop a completely
new set of strategies that rely on the fact that right adjoints preserve the
Ramsey property while left adjoints preserve the dual Ramsey property, and then
treat classes of algebras as Eilenberg-Moore categories for a (co)monad. We
first show that for any group $G$ (finite or infinite) finite $G$-sets have
finite small Ramsey degrees, and that every finite $G$-set has a finite big
Ramsey degree in the cofree $G$-set on countably many cofree generators. We
then show that finite algebras in any nontrivial variety have finite dual small
Ramsey degrees, and that every finite algebra has finite dual big Ramsey degree
in the free algebra on countably many free generators. As usual, these come as
consequences of ordered versions of the statements. To the best of our
knowledge, this is the first calculation of dual big Ramsey degrees after the
Infinite Dual Ramsey Theorem of Carlson and Simpson.
