Example

(6) The identity mapping is the counit of the cogebra A; on the cogebra A$^{(X)}$ (no. 1, Example 4) the linear form $\gamma$ such that $\gamma(e_x) = 1$ for all $x \in X$ is the counit. On the cogebra T(M) (resp. S(M), $\Lambda(M)$) the linear form $\gamma$ such that $\gamma(1) = 1$ and $\gamma(z) = 0$ for $z$ in the $T^n(M)$ (resp. $S^n(M)$, $\Lambda^n(M)$) for $n \geq 1$ is the counit. Finally, let B be an A-algebra which is a finitely generated projective A-module and has a unit element $e$; then on the cogebra B* (no. 1, Example 3) the linear form $\gamma: x^* \mapsto \langle e, x^* \rangle$ is the counit, for this form is just the transpose of the A-linear mapping $\eta_e: \xi \mapsto \xi e$ of A into B and by transposition the commutativity of diagrams (16) follows from that of the diagrams which express (using $\eta_e$) the fact that $e$ is unit element of B (§ 1, no. 3); the same argument moreover shows that conversely, if the cogebra B* admits a counit $\gamma$, the transpose of $\gamma$ defines a unit element $e = t_\gamma(1)$ of B.