Vakil (FOAG) defines the blow-up $X \hookrightarrow Y$ (closed subscheme corresponding to a finite type quasicoherent sheaf of ideals) to be Cartesian diagram
$$\require{AMScd} \begin{CD} E_X Y @>>> Bl_X Y\\ @VVV @VV{\beta}V \\ X @>>> Y \end{CD}$$
such that $E_XY$ (the scheme-theoretic pullback of $X$ by $\beta$) is an effective Cartier divisor on $Bl_XY$, such that any other such Cartesian diagram
$$\require{AMScd} \begin{CD} D @>>> W\\ @VVV @VVV \\ X @>>> Y \end{CD}$$
where $D$ is an effective Cartier divisor on $W$, factors uniquely through it:
$$\require{AMScd} \begin{CD} D @>>> W\\ @VVV @VVV \\ E_X Y @>>> Bl_X Y \\ @VVV @VVV \\ X @>>> Y \end{CD}$$
We call $Bl_XY$ the blow-up (of $Y$ along $X$, or of $Y$ with center $X$).
Next we have the exercise 22.2.A: If $U$ is an open subset of $Y$, then $Bl_{U\cap X}U \cong \beta^{-1}(U)$, where $\beta: Bl_XY\longrightarrow Y$ is the blow-up.
I think just note that the diagram
$$\require{AMScd} \begin{CD} (E_X Y)\cap\beta^{-1}(U) @>>> \beta^{-1}(U)\\ @VVV @VV{\beta|_{\beta^{-1}(U)}}V \\ U\cap X @>>> U \end{CD}$$
is the blow-up of $(U\cap X) \hookrightarrow U$. And it will be because $\beta|_{\beta^{-1}(U)}$ is the restriction of $U \subset Y$ in $\beta:Bl_{X}Y \longrightarrow Y$.
Is this right? Any tips on how to answer the exercise?