Yer free monad on a container C is yer C*, with container morphisms in I -> C* and C*;C* -> C*, giving a monad on sets.
But -* is itself a monad on containers, with container morphisms in C -> C* and...
@nprofile1q... I tend to think of these Kleisli arrows as some kind of staging in a compiler so your use of the term "template" may be similar to a C++ template, I'm not sure. Anyway, looking forward to seeing what perturbation testing looks like...