# Commuting two pullbacks

I have stumbled upon some interesting exercise whilst reading the "Category Theory for Scientists" book.

Below is the universal property of fiber products:

By using the universal property, I can try placing some set A, connect it to the X, X', Y, Y' through some functions and thus have two unique arrows spawning from it to the W and W'. However, nothing that could be used to create a direct connection between the W and W' comes out from that.

Any suggestions how this could be solved?

Thanks!

-

Consider the morphisms $W \to Y \to Y'$ and $W \to X \to X'$. Evidently they induce the same morphism $W \to Z'$. Hence, the universal property of $W'$ tells us that that they induce a morphism $W \to W'$ such that everything commutes (we don't need even need that $W$ is a pullback).
Hint: Since the square on the face behind is a pullback, look for a pair of maps $W\to Y',\ W\to X'$ to form a commuting square with $Y'\to Z'$ and $X'\to Z'$.