The 20-year-old Ghana midfielder started in an advanced role and grew into the game, going close to scoring himself before teeing up Coventry's second goal in a 2-0 pre-season win on Friday.
Manager Frank Lampard said afterwards that the club had needed to work hard to convince Yirenkyi to join, given interest from elsewhere in Europe.
"We've got a big belief in Caleb here," Lampard told Coventry City's in-house media. "He's a player we're really pleased to have brought to the club.
"He was wanted by a lot of clubs around Europe. We sold him the project. He's a good, talented, young player.
"We believe he's going to have a big career. He's a big addition to the club, and we saw signs of his qualities against Monaco, and hopefully we're going to see a lot more."
Yirenkyi's move from Nordsjælland is worth around £23.1m plus a further £2.5m in add-ons, making him Coventry's most expensive signing and the largest fee ever received by a Scandinavian club, surpassing the €25m Nordsjælland collected for Ernest Nuamah's move to Lyon in 2023.
He becomes the fourth Coventry signing to break the club's transfer record this summer, following Aurèle Amenda, Loum Tchaouna and Carl Rushworth, as Lampard's side spend heavily following promotion to the Premier League.
A product of the Right to Dream Academy, Yirenkyi made his senior Nordsjælland debut in September 2024 before being permanently promoted to the first team the following February, going on to make 56 appearances, scoring four goals and adding eight assists.
He was named the Danish Superliga's Young Player of the Year last season as Nordsjælland finished third and reached the Conference League, before a breakout World Cup with Ghana, where he scored a stoppage-time winner against Panama that helped the Black Stars reach the last 32. Coventry begin their Premier League campaign at home to Arsenal on August 21.