A polytypic function definition is a function definition that is parametrised with a datatype. It embraces a class of algorithms. As an example we define a simple polytypic "crush" combinator that can be used to calculate polytypically. The ability to define functions polytypically adds another level of flexibility in the reusability of programming idioms and in the design of libraries of interoperable components.

Each datatype constructor comes equiped not only with a so-called map and fold (catamorphism), as is widely known, but, under some condition, also with a kind of map and fold that are related to an arbitrary given monad. This result follows from the preservation of initiality under lifting from the category of algebras in a given category to a certain other category of algebras in the Kleisli category related to the monad.

this paper, we propose a new theory on monadic catamorphism bymoving Fokkinga's assumption on the monad to the condition of a map between monadic algebras so that our theory is valid for arbitrary monads including, for example, the state monad that is not allowed in Fokkinga's theory. Our theory covers Fokkinga's as a special case. Moreover, Meijer and Jeuring's informal transformation rules of monadic programs in their case study is actually an instance of our general promotion theorem.