To begin with, the House of Representatives *impeached* him before he left office. It's only the required Senate trial that's taking place after he's left office.
As for *trying* him after he has left office, there's historical precedent for impeaching and trying a federal official after they have left office. You've got an uphill climb if you want to argue that trying Trump is unconstitutional. You certainly cannot say, as the GOP has done, that it's unconstitutional on its face.
On January 15 the Congressional Research Service (CRS) produced a report entitled "The Impeachment and Trial of a Former President," that explores the history of impeachment in the United States, and notes that federal officials have been impeached after they have left office. In other words, it has been regarded as constitutional.
The CRS is a public policy research institute created by Congress in 1914. Like the Congressional Budget Office and the General Accountability Office, it works directly for Members of Congress on a bipartisan basis.
The report points out that federal officials have been impeached and tried after they have left office.
The most notable case involved the impeachment of Secretary of War William Belknap in 1876, essentially on grounds of rampant corruption. Here's what the U.S. Senate Historical Office says about it:
On March 2, 1876, just minutes before the House of Representatives was scheduled to vote on articles of impeachment, Belknap raced to the White House, handed Grant his resignation, and burst into tears.This failed to stop the House. Later that day, members voted unanimously to send the Senate five articles of impeachment, charging Belknap with "criminally disregarding his duty as Secretary of War and basely prostituting his high office to his lust for private gain."The Senate convened its trial in early April, with Belknap present, after agreeing that it retained impeachment jurisdiction over former government officials. During May, the Senate heard more than 40 witnesses, as House managers argued that Belknap should not be allowed to escape from justice simply by resigning his office.On August 1, 1876, the Senate rendered a majority vote against Belknap on all five articles. As each vote fell short of the necessary two-thirds, however, he won acquittal.
Republicans undoubtedly know about this precedent. They rely on the fact that the American public does not know about it.