The ex-Prez showed up in court Monday in NYC, where his hush-money case kicked off -- the first of his three criminal trials -- and where Donald had to sit through a day full of lawyers talking about the ins and outs of how this thing is gonna go.

Of course, they have to start with jury selection ... and as the attorneys were going back and forth, NYT reporter Maggie Haberman went on CNN to claim Trump appeared to be sleeping.

During her live-blogging of what was going on inside, Haberman wrote that Trump's head kept dropping and that his jaw seemed to go slack a few times ... signs that the guy was, perhaps, catching some ZZZs. There are other reporters who say DT's eyes were closed.

In light of this development -- which many were shocked to hear as it made its way through the media today -- a moniker that's usually assigned to Joe Biden is applying here today.

Yes, we're talking about Sleepy Don ... something the internet is having a field day with. However, as he was heading into court, Trump was alert ... and fired up talking to reporters. He called the trial an assault on the country ... and once again claimed it was a witch hunt.

Remember, this is all over alleged hush money payments Trump and his team got to Stormy Daniels to keep quiet about an affair she claims she had with him back in the 2000s. He's facing 34 counts of falsifying business records ... which is classified as a felony in NY.

A few notable items from Monday's court sesh so far ... the transcript of the "Access Hollywood" tape will be read in court, and so will deets about another alleged affair Trump had with ex-Playboy model Karen McDougal. Michael Cohen is also expected to testify.

The jury selection process is expected to take a couple weeks to get through -- as thousands of Manhattanites were summoned to be part of the process as they filter through them all.