Let $ f: Y \to X $ and $ g : Z \to X $ be two morphisms of schemes.
Suppose we know that $ Y \times_X Z $ exists, and let $ p $ dénote its projection on $ Y $, and $ q $ its projection on $ Z $.
Let $U, V $ and $ W $ be open sets in $X, Y $ and $Z$ respectively, such that $f (V) \subset U $ and $g (W) \subset U $.
Question :
How to establish that $ V \times_U W $ exists and is canonically identified with the open set $ p^{-1} (V) \bigcap q^{-1} (W) $ of $ Y \times_X Z $, combining universal properties of fiber products and universal properties of open immersions ?
Thank you in advance.
To show that $ V \times_U W $ exists, you have to find an $U$-scheme $Z$ satisfing the universal property of $ V \times_U W $. By the way, two $U$-schemes $Z,Z'$ satisfing the universal property of $ V \times_U W $ are canonically isomorphic in the category of $U$ schemes. Now, $p^{-1}(V)\cap q^{-1} (W)$ is canonically a $U$-scheme, and you can verify, as a really simple exercise, that it satisfies the universal property of $ V \times_U W $. Try it step by step, and post if blockers any.