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What role does だと play in the following sentence?

Sentence: 今、何時だと思いますか。

As far as I understand, だ means 'be; is'
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question
grammar particles particle-と embedded-question
What role does だと play in this sentence?
The copula だ and the quoting particle と.

* * *

と is used to repeat something someone has said, or what was written somewhere.
English uses quotative markers: 'He asked you: "What is a quote?"'.  
I used 'he'(pronoun) and 'ask'(a communication-related verb). Most languages
require these two constituents to introduce a quotation, and Japanese is no
exception.

Furthermore, Japanese uses と and「」(rotated for vertical text) to mark
quotations.  
_(Both と as well as the markers can be used in other ways.)_ For indirect
quotations, no 「」are used.

A quotation can be a sentence embedded into another one. If it happens that
it's a propositional sentence, it will often end with だ. And that's how you
end up with だと.

> 太郎【たろう】は「綺麗【きれい】 **だ** 」 **と** 言【い】った

'Tarou said: "It is pretty".'

と is also used with verbs of thought, just like 思【おも】います. While with 言【い】う
(verbs of saying) it was formal, for verbs of thought its use is casual.

Putting everything together, we now know that your sentence uses:

  * Indirect quotation
  * A verb of thought
  * A proposition (by someone else, possibly)

We can conclude that:

> Now, what time is it QUOTE think QUESTION
>
> "What time do you think it is now?".
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answer