# How to prove the pullback lemma

I am new in category theory. I am trying to prove the well known fact that if you have a commutative diagram of the form □□, where each square is a pullback, then the whole diagram is a pullback too, and hence deduce that the pullback of a pullback square is a pullback. Every book I have looked at has this as an exercise, but I (embarrasingly, I know) cannot see the solution. I have tried using the universality property of the two pullbacks but i am lost in calculations. If someone could help, I would really appreciate it.

• See also this question. – t.b. May 30 '12 at 13:41
• Crossposted: mathoverflow.net/questions/98378/… – Rasmus May 30 '12 at 13:49
• You might also try posting your work up to the point where you get lost in calculations; then someone might help you find your way out again. – MJD May 30 '12 at 14:18

• Thanks for a citeable reference for this proof! Trivial nitpick: $m = m' = n' \circ f$ on page 2 should be $m = m' = f \circ n'$. – darij grinberg Jun 30 '15 at 12:56