Gratefully received as it was, his British hosts still needed to pull it off, and the prime minister still needed to make it through the potential rollercoaster of a news conference with his guest.

And that is what Sir Keir managed and so hence his team's relief.

They have now - on several occasions - managed to tame Trump during their joint public appearances.

It is not that the two men agree on everything, far from it.

Their instincts, communication styles and politics are wildly different, but Trump's disagreements with Sir Keir were somehow channelled past him, rather than at him.

The UK is expected to recognise a Palestinian state in the coming days - when, from Downing Street's perspective, the president is safely back on his own side of the Atlantic. The president acknowledged to me he disagrees with the prime minister on this.

He said so explicitly, but gently, and only at the end of a lengthy answer which the Starmer would have agreed with the thrust of.

Even his remarks about illegal immigration, while headline making, seem to have less impact in this the second half of the first year of his second term. The president's willingness to comment on the internal politics of an ally feels more priced in, and so carries less shock value for many.

In advance of the news conference, there had been much speculation about the potential for his mood to sour instantly on the mention of the convicted sex offender Jeffrey Epstein.

But, confronted by it and asked in particular about Epstein's friendship with the former British Ambassador to the United States Lord Mandelson, he instantly smothered it as if with a fire blanket.

His answer was curious, claiming he didn't know Lord Mandelson - despite them meeting in the White House last week, for a start.

Downing Street may allow themselves to hope theirs is a relationship with the Trump administration that is normalising and - whisper it - at least some of the time bordering on the conventional, and so less demanding on the bandwidth and mental energy devoted to it in its early months.

The caveat, of course, with Donald Trump, is you never know.

This state visit provided the UK with invaluable face time with the president - and so the opportunity to both set out the UK's position and attempt to persuade.

The prime minister has been successful in the former, but the persuasion bit? That's rather more tricky.

Chris Mason: Delight and relief in government after Trump state visit