  In this article we defined and studied quasi-finite comodules, the cohom
functors for coalgebras over rings. linear functors between categories of
comodules are also investigated and it is proved that good enough linear
functors are nothing but a cotensor functor. Our main result of this work
characterizes equivalences between comodule categories generalizing the
Morita-Takeuchi theory to coalgebras over rings. Morita-Takeuchi contexts in
our setting is defined and investigated, a correspondence between strict
Morita-Takeuchi contexts and equivalences of comodule categories over the
involved coalgebras is obtained. Finally we proved that for coalgebras over
QF-rings Takeuchi's representation of the cohom-functor is also valid.
