I'm trying to prove the following theorem using braided diagrams:
Let $(C,\Delta,\varepsilon)$ be a finite-dimensional coalgebra. There is an algebra structure on $C^*$ given by multiplication $\Delta^*\circ \zeta$ and unit $\eta=\varepsilon^*$, where $C^*\otimes C^* \stackrel{\zeta}{\cong}\left(C\otimes C\right)^*$ is a natural isomorphism of vector spaces.
I have only problem with proving that the unit is well-defined. In the proof which I found in one article is a statement, that there is the following equality :
Why it is true ? It isn't obvious for me.