comodule

Let $(A,\mathrm{\Delta },\epsilon )$ be a coalgebra.A right $A$-comodule is a vector space $V$ with a linear map$t:V\to V\otimes A$, called the right coaction, satisfying

$$(t\otimes \mathrm{id})\circ t=(\mathrm{id}\otimes \mathrm{\Delta })\circ t,(\mathrm{id}\otimes \epsilon )\circ t=\mathrm{id}.$$(1)

An $A$-comodule is also referred to as a corepresentation of $A$.

In of commutative diagrams: