Let $K$ be a field and $C$ a non-counital conilpotent coassociative coalgebra over $K$ whose underlying $K$-vector space is finite dimensional.
Question: Can one obtain $C$ by iterately taking pushouts of trivial coalgebras (= coalgebras with zero comultiplication)? Especially, does $C$ belong to the full subcategory generated by trivial coalgebras under small colimits?
If not, can anyone give a counterexample? Is there a natural condition on $C$, which guarantees that $C$ can be obtained that way?