Let $X$ be a projective variety over field $k$ and $Y \to X$ is a map. Set $X \times X$ to be fibered product over $\operatorname{Spec}(k)$. Then $Y \times Y$ is a variety over $X \times X$ and $X$ is also variety over $X \times X$ via the diagonal map. Is it true that $$ (Y \times Y) \times_{X \times X} X \cong Y \times_X Y? $$
It seems to be true for affine varieties and then we can try to glue, but perhaps there are better/shorter proofs of this fact.
Hint: the fibre product in a category $\mathscr C$ is an object representing the functor $$T \mapsto \operatorname{Hom}(T,X)\times_{\operatorname{Hom}(T,Z)}\operatorname{Hom}(T,Y),$$ given $f:X \longrightarrow Z$ and $g:Y \longrightarrow Z$. By the Yoneda Lemma, we only need to show the required equivalence in the category of sets.