How does Yeast-two-hybrid detect interactions between several proteins in one experiment?

I am trying to understand the Y2H screening method. I can understand how we can check if two specific proteins interact with each other. For example, if we want to check whether protein A and protein B interact, we fuse A with the Activation Domain (AD) of a transcription factor and B with the Binding Domain (BD) of another transcription factor. When the two fusion proteins are transfected into a yeast cell, the reporter gene is expressed only if the AD and the BD get together and form a complete transcription factor and that happens only if proteins A and B interact with each other.

However, I have seen claims that Y2H is a high throughput system, i.e., it can be used to detect several interactions at once. But most articles online that attempt to describe it, seem vague to me.

For example, suppose I fuse the "bait" protein to the BD and several different cDNA's to the AD and let the yeast grow. Next, I observe that the reporter gene is expressed. Doesn't this observation only imply that there exists some protein from the cDNA library that interacts with our bait protein? How do we know exactly which of the target proteins have interacted with the bait and resulted in the expression of the reporter gene?