Define a pointclass to be:

• boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions, and

• lightface inductive-like if it is $\omega$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions.

Given a lightface inductive-like pointclass $\Gamma$ we can define the boldface inductive-like pointclass ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$ as usual (this does not have much to do with the particulars of "inductive-like.")

My question is, can we do the reverse? That is, given a boldface inductive-like pointclass ${\bf \Gamma}$ can we find a lightface inductive-like pointclass $\Gamma$ such that ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$? (We wouldn't expect uniqueness; if $\Gamma$ works then so does $\Gamma(z)$ for any real $z$.)

I have heard that Howard Becker proved something like this, but I can't find it and I don't know what kind of pointclasses it was for (probably more general than "inductive-like," which is merely the case I happen to be interested in at the moment.)