Ashers Bakery are challenging the ruling over their decision in 2014 not to make a cake iced with the slogan "Support Gay Marriage".
"The notion that Christians may exercise their faith on Sundays but forget about it when they step into work on Monday is not real freedom of religion and is not freedom of conscience", Scoffield told the court.
"The result of the approach taken by the district judge and court of appeal ... is that someone providing bespoke goods must provide goods at the request of the customer, provided that is not unlawful - however offensive".
Scoffield added that the case raised an issue of principle since those with deeply held religious beliefs could be compelled to act against them.
He said the McArthur family were being forced to use their skills, trade and experience for a goal inconsistent with their beliefs and claimed they must "choose between operating their businesses or living and acting in accordance with their religious beliefs, and we say that can not be the law".
He said to create the campaign cake was to promote same-sex marriage and stated: "This is a case of forced or compelled speech".
The bakery had said its disagreement was not with the customer, but with the message requested on the cake.
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The firm argued that the cake's message was against the bakers' religious views.
Lee's order was made shortly after the Democratic Unionist party used its power of veto in the Northern Ireland assembly to block moves to make same-sex marriage legal in the country.
"Although I strongly disagree with Ashers stance against gay marriage, that is their right in a free and democratic society", Tatchell said.
Judges from the London-based Supreme Court heard the bakery's appeal at a special sitting in Belfast that is due to continue Wednesday.
Now, following a reference from the Attorney General for Northern Ireland, it's time for the Supreme Court to have its say.
This was backed up by the District Court and the High Court.
The company was ordered to pay 500 pounds sterling (about $680) compensation to Lee in 2015, after a Belfast court ruled the bakery had unlawfully discriminated against Lee on the grounds of sexual orientation and religious belief or political opinion.