I have a question about comultiplication for coalgebras:
Suppose $C$ is a coalgebra over the field $k$. How does one show that the comultiplication map $\Delta:C\to C\otimes C$ is a coalgebra map if and only if $C$ is cocommutative?
The main problem that I encounter is that when I tried to do it by definition, I was struggling to find a relation between $c_{(1)(2)}$ and $c_{(2)(1)}$. In that case, is there any other way for which I could tackle this question?