Re: Naturality Squares and Pullbacks

Natural transformations for which the naturality square is a pullback are commonly called cartesian: it's not an ideal name for them, but it's quite well established. For the question of when the unit and multiplication of a monad on Sets are cartesian, see section 3 of "Connected limits, familial representability and Artin glueing" by A. Carboni and P.T. Johnstone (Math. Struct. Comp. Sci. 5 (1995), 441--459). Peter Johnstone