Problem:

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?

Question: How do we know exactly which of the target proteins have interacted with the bait and resulted in the expression of the reporter gene?

Please explain how the targets results.