counit
An algebra has a unit: a special element 1 that does nothing when you multiply by it. Dualizing, a coalgebra has a counit: a special linear functional ε that does nothing — in the appropriate sense — when you comultiply and then evaluate one factor. It is the 'do-nothing observable' that the splitting operation must respect.
Formally the counit of a coalgebra C is a linear map ε : C -> k satisfying (ε ⊗ id) ∘ Δ = id = (id ⊗ ε) ∘ Δ, where we identify k ⊗ C ≅ C ≅ C ⊗ k. In Sweedler notation, writing Δ(c) = Σ c(1) ⊗ c(2), the axiom reads Σ ε(c(1)) c(2) = c = Σ c(1) ε(c(2)). So applying ε to one leg of the coproduct and contracting reconstructs the original element.
The duality to the unit is exact. If A is a finite-dimensional algebra with unit map u : k -> A, 1 ↦ u(1), then the dual map u* : A* -> k is precisely the counit of the dual coalgebra A*. For the function algebra k[G] on a group, the counit is evaluation at the identity element: ε(f) = f(e).
In the group algebra k[G], elements are sums Σ a_g g, comultiplication is Δ(g) = g ⊗ g, and the counit is ε(g) = 1 for every g, extended linearly. Then (ε ⊗ id)Δ(g) = 1 · g = g, as required.
The counit of a group algebra sends every group element to 1.