Naturality Squares and Pullbacks

A natural transformation is an indexed family of arrows such that a certain diagram commutes. One could require a stronger condition, namely that the said diagram is a pullback. What would such a transformation be called? I'm sure I've seen this in the literature before but I cant remember where. Pointers? This problem arose in the context of finitary monads where T(X) is the derived operations over a set X for some signature. The naturality square for the unit turns out to be a pullback. This then implies that the unit of the monad is a monic - presumably this is a result in the literature somewhere. Again, pointers? Neil Ghani