homological algebra - Comodule exercises desired

My favorite exercise is: prove that a comodule (over a coalgebra over a field) is coflat if and only if it is injective. This presumes that you already know that any coalgebra is the union of its finite-dimensional subcoalgebras, and any comodule is the union of its finite-dimensional subcomodules (if you don't know that, prove that first).

Also: construct a counterexample showing that infinite products in the category of comodules are not exact in general. Describe as explicitly as possible the cofree coalgebra with one cogenerator, i.e. the coalgebra $F$ such that coalgebra morphisms from any coalgebra $C$ into $F$ correspond bijectively to linear functions on $C$. Prove existence of the cofree coalgebra with any vector space of cogenerators.

Define a cosimple coalgebra as a coalgebra having no nonzero proper subcoalgebras, and a cosemisimple coalgebra as a direct sum of cosimple coalgebras. Prove that a coalgebra is cosemisimple if and only if the abelian category of comodules over it is semisimple. Prove that any coalgebra $C$ contains a maximal cosemisimple subcoalgebra which contains any other cosemisimple subcoalgebra of this coalgebra. Consider the quotient coalgebra of $C$ by this maximal cosemisimple subcoalgebra; it will be a coalgebra without counit. Prove that any element of this quotient coalgebra $D$ is annihilated by the iterated comultiplication map $Dto D^{otimes n}$ for a large enough $n$ (depending on the element).

Define Cotor between an unbounded complex of right comodules and an unbounded complex of left comodules as the cohomology of the total complex of the cobar complex of the coalgebra with coefficients in these two complexes of comodules, the total complex being constructed by taking infinite direct sums along the diagonal planes. Construct a counterexample showing that the Cotor between two acyclic complexes can be nonzero.