There doesn't seem to be any mention of something like a comagma: the categorical dual of a magma where the arrows are reversed for • : A × A → A, as in π : A → A × A. Only coalgebras and bialgebras are mentioned, but they have extra requirements for identity and associativity.
If comagmas exist, then a bimagma could be constructed from a set S with (S, •, π) where π(•(x,y)) = (x,y) and •(π(x,y)) = (x,y). This is the same structure as a Jónsson-Tarski algebra.
Are these valid assumptions, and are comagma and bimagma known structures that have been studied?