Composition of Cartesian product is Cartesian

Question

Let $X_1,X_2, X_3$ and $Y$ be schemes over $\mathbb{C}$. Let $f_1:X_1 \to Y, f_2:X_2 \to Y$ and $f_3:X_3 \to X_2$ be three morphism of schemes. Under the composition map $f_2 \circ f_3$ from $X_3$ to $Y$, is it true that the Cartesian product $X_3 \times_Y X_1$ is isomorphic to $X_3 \times_{X_2} (X_1 \times_Y X_2)$?

Answer 1

Yes, indeed it is a property of categorical pullback in any category. You can construct "towers" of pullbacks and still get a pullback. More precisely, if you have a commutative diagram

where the right hand square is a pullback, then: the left hand square is a pullback if and only if the big square is a pullback.