Let $R$ be a commutative ring, $C$ a coalgebra over $R$. I am asking about the category of $C$-comodules $C$-Comod.
It is clear that if $C$ is a flat $R$-module, then $C$-Comod is abelian. Hence, is my first question.
Minor Question: Is the flatness a necessary condition for $C$-Comod to be abelian?
The flatness is useful because it ensures that a kernel of a map is a subcomodule. Hence, it is suggestive to declare all injections admissible monomorphisms and all surjections, whose kernel is a subcomodule, admissible epimorphisms.
Main Question: Does this define a structure of an exact category on C-comod?
PS I hope the answer is yes but I am afraid it is no. I do not see how to ensure existence of the pull-back $X\times_YZ$ where $X\rightarrow Y$ is an admissible epimorphism. Doesn't it need a kernel?
