Let $k$ be a field. Let $C$ be a coassociative and counital coalgebra over $k$. Takeuchi defines the notion of quasi-finite comodule as follows: a left $C$-comodule $M$ is quasi-finite if the induced functor $M\otimes -:Vect_k \rightarrow LComod_C$ is a right adjoint.
Examples of quasi-finite comodules are given by finitely cogenerated comodules: this means $M$ is isomorphic to a subcomodule of the cofree comodule $C\otimes V$ where $V$ is a finite dimensional vector space.
Takeuchi says that not every quasi-finite comodules are finitely cogenerated. What are examples of such comodules?