A coalgebra is a triple $(A,\Delta,\epsilon)$ consisting of a vector space, a coproduct, and a counit. Now as we all know, just like the unit in an algebra, the counit of a coalgebra is unique, i.e. if there exists another $\epsilon'$ satisfying the counit axiom, then since
$$
\epsilon(a) = \epsilon(a\_{(1)}\epsilon'(a\_{(2)})) = \epsilon'(\epsilon(a\_{(1)})a\_{(2)})=\epsilon'(a).
$$
So if there is no "choice" of counit, would it not be better define a coalgebra to be a coassociative pair $(A,\Delta)$ for which there exists a counit. This is how we define a unital algebra and a group, so why the different approach for coalgebras?



What is a coalgebra?
Is it how we define a unital algebra? I’d define a unital algebra as coming endowed with a unit, rather than just asserting one exists. For one reason, then the natural notion of map is a unital map. But at any rate there’s just no distinction here between the definitions algebras and coalgebras, people who define algebras using a unit property will do so for coalgebras too, and people who define algebras using a unit structure will do so for coalgebras too.


