Proposition 8

— Let M be a free A-module, then with its algebra and cogebra structures, TS (M) is a graded commutative and cocommutative bigebra. The counit is the A-linear mapping $\varepsilon : TS(M) \to TS^0(M) = A$ which is zero on $TS^p(M)$ for $p > 0$ and such that $\varepsilon(1) = 1$.

We know that the A-algebra $TS(M)$ is associative, commutative and unital. On the other hand, the coproduct is by construction a homomorphism of graded algebras; now the fact that the cogebra $TS(M)$ is coassociative and cocommutative follows by an easy calculation from the Formula (10). The mapping $\varepsilon$ of $TS(M)$ into A is a homomorphism of graded algebras such that $\varepsilon(l) = l$. Finally for every $x \in M$ we have $\varepsilon(\gamma_p(x)) = 0$ if $p > 0$, $\varepsilon(\gamma_0(x)) = l$; if we bear in mind (lo), this shows that $(F \otimes 1) \circ c = (l \otimes E) \circ c = \mathrm{Id}_{TS(M)}$; thus $E$ is the counit of $TS(M)$.