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Take a look at this link.

7. After all the thinking hats have had their say, the blue hat continues to transition between the hats in a logical order. It may, for instance, ask the red hat for its intuitive insights about the green hat’s ideas. Or, it may ask the white hat to gather more facts and information about the dangers that the black hat brought to light. After which, it may ask the yellow hat to bring forth some logical solutions to the problem at hand.

Can I summarize the above in the following way:

When all of the hats are done with giving their opinions, it is again blue hat's turn to invite cross-talks between various hat-pairs.

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The example is a little cryptic. Is this supposed to be literal hats that can talk? Or is, for example, "blue hat" referring to a person wearing a blue hat? Well, whatever, I guess it's not relevant to the question.

Short answer: No, that is not a generally recognized definition of the term "cross-talk". I'm not sure that there is a generally recognized definition of "cross-talk". It's a term used by people who work with radios and telephones to refer to interference between different signals, but that's clearly not at all what you mean here. You appear to be using it to mean something more like "productive conversation and sharing of ideas".

On a side note, I'm not sure why you say "between various hat-pairs", as opposed to simply "between the hats". That implies that, say, red can talk to blue and blue can talk to black, but read and blue cannot simultaneously listen to what black has to say, and then red reply while blue and black listen. I'm not sure if that's what you want to say or not.

I'd probably say, "It is again blue hat's turn to invite discussion between the hats." Or instead of "discussion" we might say "conversation" or "sharing of ideas".

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"Crosstalk" is uncountable and so shouldn't be pluralized. It is also not hyphenated. It is usually considered an undesirable thing, something that interferes with communication by not keeping channels separate.

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