  Let $R$, $S$ be two rings, $C$ an $R$-coring and ${}_{R}^C{\mathcal M}$ the
category of left $C$-comodules. The category ${\bf Rep}\, ( {}_{R}^C{\mathcal
M}, {}_{S}{\mathcal M} )$ of all representable functors ${}_{R}^C{\mathcal M}
\to {}_{S}{\mathcal M}$ is shown to be equivalent to the opposite of the
category ${}_{R}^C{\mathcal M}_S$. For $U$ an $(S,R)$-bimodule we give
necessary and sufficient conditions for the induction functor $U\otimes_R - :
{}_{R}^C\mathcal{M} \to {}_{S}\mathcal{M}$ to be: a representable functor, an
equivalence of categories, a separable or a Frobenius functor. The latter
results generalize and unify the classical theorems of Morita for categories of
modules over rings and the more recent theorems obtained by Brezinski,
Caenepeel et al. for categories of comodules over corings.
