If \(\mathcal C\) is a category and \(F\colon\mathcal C\to\mathcal C\) is an endofunctor, then an \(F\)-coalgebra is a pair \((C,\alpha)\), where \(C\) is an object of \(\mathcal C\) and \(\alpha\colon C\to FC\) is a morphism. An \(F\)-coalgebra morphism \(f\colon(C,\alpha)\to(C',\alpha')\) is a \(\mathcal C\)-morphism \(f\colon C\to C'\) such that \((Ff)\alpha=\alpha'f\). We thus have a category \(\text{Coalg\,}F\) of \(F\)-coalgebras.
The author explains how the categories of \(R\)-coalgebras (\(R\) a commutative ring), of \(A\)-corings (\(A\) an \(R\)-algebra which is not necessarily commutative), and their respective categories of comodules can be defined as full subcategories of \(\text{Coalg\,}F\) for suitable functors \(F\). The main theme of the paper is to explain how properties of categories of coalgebras, corings and comodules can be obtained directly from well known results of categorical algebra by simple dualization. New results concerning the existence of limits and of factorizations of morphisms are obtained.