Let $X$ be a separated scheme, $U,V \subseteq X$ open affine subschemes, $\Delta \colon X \to X \times X$ the diagonal morphism and $\pi_1, \pi_2 \colon X \times X \to X $ the natural projections, so that $\pi_i \circ \Delta = \text{id}_X$ for $i=1,2$.
It is easy to see by checking the universal property that $\pi_1^{-1}(U) \cap \pi_2^{-1}(V) \cong U \times V$. My question is, do we have $$ \bigl|\Delta^{-1}\bigl(\underbrace{\pi_1^{-1}(U) \cap \pi_2^{-1}(V)}_{\cong \, U \times V}\bigr)\bigr| = U \cap V$$ as sets, as we would expect from the situation in the category of sets? Notice that I'm not talking about bijection but equality of sets.