A coalgebra $C$ is called cosimple if it has no subcoalgebras. It is called cosemisimple if it is a direct sum of simple coalgebras.
Is this direct sum decomposition unique? Explicitly, can there exist another decomposition into simple coalgebras which are of different isomorphic class or appear with different multiplicities?
Moreover, n-lab says that cosemisimplicity
is equivalent to saying that every $C$-comodule is a direct sum of simple subcomodules.
Why is this true?
