It's not hard to show that if $X$ is uncountable Polish and $\Gamma$ is a self-dual pointclass closed under continuous preimages, then there is no $X$-universal set for $\Gamma(X)$ with a diagonalization trick, this is exercise 22.7 in Kechris' book, so in particular there is no $X$-universal set for any $\mathbf{\Delta}^0_\xi(X)$.
According to this question if $X,Y$ are uncountable Polish spaces then there is no $Y$-universal set for $\mathbf{\Delta}^0_\xi(X)$, where that means a set $U\in\mathbf{\Delta}^0_\xi(Y\times X)$ such that $\mathbf{\Delta}^0_\xi(X)=\{U_y\mid y\in Y\}$, where $U_y=\{x\in X\mid (y,x)\in U\}$ are the slices of $U$. I don't see a simple way to prove this generalization of the result and I'm having troubles finding a proof in the literature.
How can the above result be proved? And does it extend to other self dual pointclasses closed under continuous images?