I found out that pullback are stable under retractions (right invertible maps). So that's kind of the opposite of what I was looking for. Well, I maybe I made a mistake in the first place and that is actually what I want.
I know that pullbacks are stable under monomorphisms. I have a case where f: A -> B is not only a mono, but even a left invertible, i.e., there exists an l: B -> A with l \circ f = id_A. My question is whether pullbacks are also stable in this case, i.e., whether the corresponding projection in the pullback diagram is also left invertible.
I found out that pullback are stable under retractions (right invertible maps). So that's kind of the opposite of what I was looking for. Well, I maybe I made a mistake in the first place and that is actually what I want.