I am trying to show that every coalgebra is the quotient of a cosemisimple coalgebra (I'm actually not sure if it's true or not).
Here is my attempted solution:
Let $ C $ be a coalgebra. We know that $ C = \sum_i C_i $ is the sum of its finite dimensional subcoalgebras. Let $ n_i = [C_i : k] $, then for each $ i $ there is an onto coalgebra map $ f_i: C_{n_i}(k) \to C_i $, where $ C_{n_i}(k) $ denotes the comatrix coalgebra of degree $ n_i $. Hence there is an onto coalgebra map $ f: \bigoplus_i C_{n_i}(k) \to C $. Now each $ C_{n_i}(k) $ is simple, hence $ \bigoplus_i C_{n_i}(k) $ is cosemisimple, and we are done.
Is this solution correct?