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.
Just for you, and it turns out my answer has to contain at least 30 characters, so let's make it 100.
My advice: try once more. The pullback property for the big square (or rectangle) really follows from the pullback properties of the two small squares.
Hint: Start with the right hand square. Can you see two arrows as in the condition of the pullback property?