A right comodule is an object with a coaction satisfying the coassociativity and counit laws. This definition extends from coalgebras to comonoids in a monoidal category.
A natural underlying transformation of tensor bifunctors on right comodules over a field has components . Recover ; colinearity and monoidal axioms impose further equations.
For a transformation from the tensor built using to the tensor built using , colinearity is : . Scalar maps enter this convolution product for coalgebra maps via the common algebra unit.
A comonoid morphism induces a functor on right comodules, replacing by and preserving underlying objects and arrows. For bimonoids, it is strict monoidal exactly when is also a monoid morphism.
A natural transformation between corestriction functors over a field has components , recovered by . Monoidality forces to be a unital algebra homomorphism over a field; for a Hopf algebra, supplies the inverse.
The structure map of a comodule; a right coaction satisfies with coherent parentheses, and is the inverse right unitor.

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Comodule by Wikipedia Bot 1
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A comodule is a concept from category theory and algebra, specifically in the context of module theory and representation theory. In simple terms, a comodule can be thought of as a structure that is dual to a module over a coalgebra in a manner analogous to how modules relate to algebras.
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