Idah, 24, arrived at the Swansea.com Stadium in the recent transfer window in a £6m deal, becoming the club’s record signing outside the Premier League.
The Republic of Ireland international had only joined Celtic permanently for £9m last summer after a productive loan spell, scoring 20 goals last season.
Yet despite that return, he was allowed to move on after struggling to secure a consistent starting role in Glasgow.
For Sheehan, Swansea’s ability to land a player from one of Britain’s biggest clubs speaks volumes about the foundations now being rebuilt.
“Celtic are a huge football club and the expectation on that football club is massive. So for us to have the opportunity to present to him, to show what we are and what we want to be and where he fits into it, makes it a great signing for us,” said the Swans head coach at the club's training ground.
“I am really happy to have him here and we will protect him, we will look after him, we will get him to a place where he is good.
“I say that to all the new signings coming in - it is not how you come into a club, it is how you leave a football club.
“This is the culture, this is what we do, these are the fundamentals of how we work every day, this is how you fit into it and this is how we move forward as a team.”
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Idah himself has described the move as a “fresh start” and, according to Sheehan, is eager to back up his words with performances on the pitch.
“I think that was one of the things that he said - he wants to prove it,” Sheehan explained.
“That’s good. We don’t want people that want to talk about doing stuff, we want people that are going to do it.”
Swansea had pursued a centre-forward all summer, exploring moves for former striker Oli McBurnie as well as Richard Kone and others.
Landing Idah, a player Sheehan and his staff had long admired, was seen as a coup.
“I think everybody knew that we wanted to strengthen our nine competition,” Sheehan said.
“So to bring in somebody of his calibre, his physicality, his athleticism, his mentality to come here – we were delighted with that.
“He’s a player that I know a lot about, my coaching staff know a lot about, so when the opportunity came to have conversations it was a no-brainer really and credit to the ownership that fully backed that.”
Idah trained with his new teammates for the first time on Thursday after returning from international duty with Ireland, and Sheehan admitted the impact was immediate.
“You could feel that reaction,” he said of Idah’s first session.
Whether the new signing starts against Hull City on Saturday remains to be seen.
Slovenian striker Zan Vipotnik has three goals in as many league games and added another for his country against Sweden during the break.
“I’m delighted he’s still here,” Sheehan said of Vipotnik, who had been linked with a late move away.
“He’s had a great week and now he’s showing what he’s capable of. He’s added that grit and that aggression that’s not been consistently there.
“We want players here that have a point to prove, who want to hold on to the shirt, they want to play for the shirt and take us up a level.”
