Former President Trump has arrived at the courthouse for today's proceedings.

This will be the final day of court this week, as court is off on Friday for Trump to attend his son Barron's high school graduation.

Michael Cohen has arrived at the lower Manhattan courthouse for his testimony this morning.

Proceedings are scheduled to get underway at 9:30 a.m. ET.

Testimony is scheduled to conclude at 4 p.m., after which Judge Juan Merchan is scheduled to hear arguments from the parties about the bounds of potential testimony from a defense expert witness.

Michael Cohen returns to the witness stand this morning, where the former Trump attorney is expected to face a full day of cross-examination.

Cohen, under direct examination earlier this week, described in-person meetings and phone calls with Donald Trump, who he said joined into an agreement with tabloid publisher David Pecker to catch and kill negative stories ahead of the 2016 election; approved a $130,000 hush money payment from Cohen to Stormy Daniels; and signed off on an arrangement to reimburse Cohen in 2017 using what prosecutors say were falsified invoices. Trump has denied all wrongdoing.

Defense attorneys are expected to question Cohen's credibility based on his past testimony and previous statements he has made in interviews, podcast and books.

Former President Trump has asked New York's highest court to rescind the limited gag order that prevents him from commenting publicly about witnesses, jurors and lawyers in his ongoing criminal trial, according to a new court filing.

The filing is sealed, but is the next step after an intermediate appeals court yesterday upheld the order imposed by trial Judge Juan Merchan.

Trump has frequently attacked the judge, which is permitted, and has called the limited gag order unconstitutional.

In its ruling yesterday, the intermediate appellate court said that Merchan had appropriately balanced Trump's free speech rights with the court's need to control the trial.

There was no immediate comment from the Manhattan DA's office.

-ABC News' Katherine Faulders contributed to this report

Jurors briefly saw text messages between Cohen and Trump staffer Keith Schiller on June 16, 2016, in which Cohen said he was trying to contact Trump through Schiller to update him on the Karen McDougal situation.

"Where's the boss?" Cohen texted.

Jurors then saw text messages between Cohen and National Enquirer Editor Dylan Howard, who met McDougal in person to vet her allegations on June 20, 2016.

"Understood, I got this locked down for you, I won't let it out of my grasp," Howard texted Cohen.

Cohen said he later had a call with Howard and National Enquirer publisher David Pecker to learn about the McDougal meeting.

"That they believed that they had this control, as Dylan had stated to me," Cohen said about what he learned on the call.

In court, the jury is focused as Cohen recounts the catch and kill efforts. Earlier, some jurors had laughed and flashed a smile at Cohen's jokes, but they have now returned to their familiar serious faces. Many appear to be taking notes and they're often looking at Cohen and prosecutor Susan Hoffinger as she questions him.