Generator coalgebras are not necessarily quasi-coFrobenius

We study the problem of whether a coalgebra that generates its category of left (right) comodules is left (right) quasi-coFrobenius or not. We prove it does not hold in general, by giving a method of constructing counterexamples. This gives a negative answer to a question stated in \cite{kn:coalgen}. We also prove it is true for monomial pointed coalgebras and we characterize the quivers $Q$ such that $\mathbb{k}Q$ admits a monomial subcoalgebra that is left (right) quasi-coFrobenius.