Are cofree comodules Quasi-finite?

Are cofree comodule quasi-finite, where by quasi-finite I mean there is a left adjoint to the cotensor functor?

No, let $C$ be an $R$-coalgebra, the $C$-comodule $\underset{n\in \mathbb{N}}{\bigoplus} C$ is cofree but not quasifinite, by SAFT.