In the semantics of programming languages one can view programs as state transformers, or as predicate transformers. Recently the author has introduced state-and-effect triangles which capture this situation categorically, involving an adjunction between state- and predicate-transformers. The current paper exploits a classical result in category theory, part of Jon Beck's monadicity theorem, to systematically construct such a state-and-effect triangle from an adjunction. The power of this construction is illustrated in many examples, covering many monads occurring in program semantics, including (probabilistic) power domains.

• 18A40 - Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
• 68N15 - Theory of programming languages
• 68N30 - Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.)
• 68Q55 - Semantics in the theory of computing