Comodules, contramodules and Pontryagin duality

Let $C$ be a $k$-coalgebra, where $k$ is a field. The category of pseudocompact left $C^{\ast }$-modules is dual to both the category of discrete right $C^{\ast }$-modules and to the category of left $C$-comodules. We obtain this way two sides of a square of dualities. In this note we complete the square by introducing as fourth corner the category of pseudocompact right $C$-contramodules. We use these dualities to extend a natural isomorphism of Takeuchi.