  Let $C$ be a $k$-coalgebra, where $k$ is a field. The category of
pseudocompact left $C^*$-modules is dual to both the category of discrete right
$C^*$-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.
