Flexible presentations of graded monads

A large class of monads used to model computational effects have natural presentations by operations and equations, for example, the list monad can be presented a constant binary operation subject unitality associativity. Graded are generalization that enable us track quantitative information about being modelled. Correspondingly, graded using an existing notion presentation. However, has some deficiencies, in particular many do not presentations. We introduce flexibly presentation does suffer from these issues, develop associated theory. show every induces equipped with interpretations presentation, all satisfying condition on colimits As part this, we usual algebra-preserving correspondence between transfers monads.