Let $(A, M, u)$ be a finite dimensional algebra where $M: A\otimes A \rightarrow A$ denotes multiplication and $u: k \rightarrow A$ denotes unit.
I want to prove that $(A^*, \Delta, \varepsilon) $ is a colagebra where $\Delta: A^*\rightarrow A^* \otimes A^*$ is a composition: $A^* \overset{M^*}{\rightarrow}(A\otimes A)^* \overset{\rho^{-1}}{\rightarrow}A^*\otimes A^*$
And $\rho: V^*\otimes W^* \rightarrow (V\otimes W)^*$ is given by $<\rho(v^*, w^*), v\otimes w>=
I have proven that $\rho$ is injective and since $A$ is finite dimensional $\rho$ is also bijective and we can take the inverse $\rho^{-1}$.
But I have problems understanding how does $\Delta$ work.
By definition we have $
P.S. Please correct me if I have grammar mistakes. Thanks!