This paper describes a theory of co-design. The objects of investigation are “design problems”, defined as tuples of “function space”, “implementation space”, and “resources space”, along with the relations that relates implementation to function and implementation to resource usage. “Co-design problems” are defined as the composition of design problems by three operations, roughly equivalent to the concepts of series, parallel, and feedback. “Monotone” design problems are those for which functions are partially ordered and there is an order-preserving map between functions and resources. The main results is that any composition of monotone co-design problem is monotone, and that there exists a systematic procedure to obtain the solution to the composed problem explicitly from the solution of the primitive problems.