The classical notion of algebras in the context of k-vector spaces or R-modules generalize in a natural way to a suitable notion of `monad’ in a higher-dimensional categorical structure, that of a double category D. Starting from the base case of a category equipped with a monoidal structure, we will discuss monoids and their dual notion of comonoids therein, and we will extend a folklore result that enriches the space of algebra maps with a comultiplication (using the so-called Sweedler’s `universal measuring coalgebras’) to the double categorical context. Such a process opens paths for further applications involving operads and other related structures, within a general framework that may be called `enriched duality’.