By Jason Farrell, home editor
Is this just a lavish holiday for Mr Trump?
At last night's banquet at Buckingham Palace the special relationship was gilded in pageantry, but today the cracks may quickly reappear, as a divisive president continues to tread, not too carefully, into fragile British politics.
His exchange of insults with London Mayor Sadiq Khan has already set the tone for his relationship with the Labour party. Trump tweeted that the mayor was a "stone cold loser".
Today, Labour leader Jeremy Corbyn will give his view of the president – and the fact that he's is giving it at a rally of protesters, campaigning against the presidential visit gives you an indication of what he might say.
Stephen Doughty, Labour MP for Cardiff South and Penarth, summed up the mood of many opposition MPs. He told Sky News: "The prime minister should never have offered this to president Trump.
"He's clearly shown that he's unwilling to play by the normal standards of diplomacy when it comes to the mayor of London and I think it's hugely ill-judged.
"I think we will regret having him here. I think it's a week of shame for Britain that he's actually come here and been given this red-carpet treatment."
Against this backdrop Mr Trump and Theresa May will be trying to do business – meeting with senior UK and US business leaders. Trade will be top of the agenda.
Mr Trump says the UK needs to slip its EU "shackles" for a US trade deal, but with Britain in a state of flux over Brexit it's hard to see where any progress can be made.
On Monday night the president tweeted: "London part of trip is going really well. The Queen and the entire Royal Family have been fantastic.
"The relationship with the United Kingdom is very strong. Tremendous crowds of well wishers and people that love our country.
"Haven't seen any protests yet, but I'm sure the Fake News will be working hard to find them. Great love all around.
"Also, big Trade Deal is possible once UK gets rid of the shackles. Already starting to talk!"
Controversially, US ambassador Woody Johnson indicated the US will want access to the NHS business in any post-Brexit trade deal.
This adds to concerns that the UK may be obliged to accept chlorinated poultry as part of any trade deal. But business leaders believe progress can be made.
Ben Digby, international director at the CBI said: "Free trade agreements always take a number of years. I think we've had some experience of that in the last couple of years in the Brexit negotiations – but there are a few things that we could be getting on with immediately to facilitate trade between both countries.
"I'd look at people – the visa system. Let's look at making it a bit easier for people to travel and do business in each country."
After meeting business leaders the president will visit 10 Downing Street for lunch. Mrs May is set to raise the issue of climate change.
A spokesman said "As the prime minister has said previously, we were disappointed by the US decision to withdraw from the Paris Agreement in 2017 and continue to hope they will return."
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