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Compatibility means that the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, that the multiplication and the unit of the algebra both be coalgebra morphisms: these statements are equivalent in that they are expressed by the same diagrams. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism.
As reflected in the symmetry of the diagrams, the definition of bialgebra is self-dual, so if one can define a dual of B (which is always possible if B is finite-dimensional), then it is automatically a bialgebra.