The McArthur family is challenging a ruling that it broke the law by refusing to decorate a cake with the slogan 'support gay marriage'.
The owners of Ashers Bakery in County Antrim have argued that completing the order would have been incompatible with their religious beliefs.
Their case, which will be streamed live online, shall be heard during a two-day hearing in Belfast. A judgement is expected last this year or in early 2019.
Ciaran Kelly from the Christian Institute, which is support Ashers Bakery, told Premier: "Should they be compelled to support a campaign or an idea with which they so fundamentally disagree? We say 'no'... there are big implications if the court rules against Ashers.
"Let's say there's a Muslim printer, and someone comes and asks him to produce something with a cartoon of the prophet Muhammad on it. We would see that there wouldn't be grounds for him to turn that order down."
Gareth Lee, a member of the Queerspace gay rights advocacy group placed an order for the cake at Ashers on 9th May 2014 but was informed two days later that it was being cancelled. He was offered a full refund.
One year later, a district judge sided with Northern Ireland's Equality Commission by ruling Ashers had acted unlawfully and directly discriminated against Mr Lee on grounds of his sexual orientation and political beliefs.
Ashers launched an unsuccessful appeal Belfast's Royal Courts of Justice in 2016 before announcing last year an intention to appeal to the Supreme Court, the UK's most senior court.
Click here to listen to Premier's John Pantry and Rosie Wright speaking with Ciaran Kelly at the Christian Institute:
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