Does anyone know of a reference for the fact that if $\lambda$ is a limit of Woodin cardinals, then there is a universal $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ set of reals?

This seems not entirely trivial to me. Suppose that $A \in (\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$, say $$x \in A \iff \exists B \in \text{Hom}_{\mathord{<}\lambda}\,(HC; \in, A) \models \varphi[x].$$ It seems like we need to bound the complexity of the formulas $\varphi$ we are considering. One way to do this is to show that if there is a set $B \in \text{Hom}_{\mathord{<}\lambda}$ with $(HC; \in, B) \models \varphi[x]$ then there is also a set $B' \in \text{Hom}_{\mathord{<}\lambda}$ with $(HC; \in, B') \models \psi[x]$, where $\psi$ is the Skolem normal form of $\varphi$. We can get $B'$ using the fact that $\text{Hom}_{\mathord{<}\lambda}$ is projectively closed and every $\text{Hom}_{\mathord{<}\lambda}$ set admits a $\text{Hom}_{\mathord{<}\lambda}$ uniformization.

These existence of these universal sets (or the existence, under $\mathsf{AD}^+$, of a universal $\Sigma^2_1$ set of reals, which seems like essentially the same thing) gets used a fair amount, but I don't recall having seen this argument (on any other argument for it) anywhere. Is there a simpler argument, or a published reference for this one?