What role does だと play in the following sentence?
As far as I understand, だ means 'be; is'
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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.
'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:
We can conclude that:
Now, what time is it QUOTE think QUESTION
"What time do you think it is now?".