The 2-monad (-)^2 on CAT has unit I described by identities, and multiplication C described by composition. For a category K, a functor F : K2 → K satisfying FI = 1 admits a unique, (normal) pseudo-algebra structure for (-)2 if and only if there is any natural isomorphism F(F2) → FC. When this is the case, the unique coherent natural isomorphism satisfies a simple equation. Moreover, the set of all natural transformations F(F2)→ FC forms a commutative monoid isomorphic to the centre of K.
As Korostenski and Tholen showed, such `factorization algebras' are of interest as they correspond to factorization systems on K.
This is joint work with Richard Wood