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∗
Nearby in the stack
-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.