With Deila's assistant, though, he is a bit more familiar. As a left-footed, free kick taker who grew up supporting Celtic in the 1990s, it probably is not all that surprising to learn that John Collins used to be one of Mulgrew's heroes. The prospect, then, of learning from the new Celtic No.2 on the training ground every day is an enticing one for the 28 year-old.
"When I grew up John was the player I looked up to, to be honest," said Mulgrew. "He wore No.11 for Celtic and he scored goals at Ibrox. He was one of my heroes growing up. I remember him scoring a free kick at Ibrox, when there was Celtic fans there, and he curled it low round the wall. He was a great player, a great servant for Celtic and I'm sure everyone is delighted to have him back."
There may be a new management duo taking pre-season training but the objectives remain the same. Celtic's season will hinge on whether they can reach the group stage of the Champions League for a third successive year but to do so they will need to navigate their way through three qualifying rounds first, the first of which commences in just over a fortnight's time. That gives little time to prepare and get up to speed, but at least the Celtic players have been over this course a few times before.
"That gives you experience and you know what you're up against," acknowledged Mulgrew. "Last year against Elfsborg they were into their [domestic] season and you could see playing the game how much they wanted to be in the Champions League, understandably so.
"So you need to be prepared for it and we'll need to get up to sharpness as quickly as possible.
"We've [got Champions League pedigree] and that's good as well. We don't want to put ourselves under too much pressure but that's where everybody at the club wants to be. We've been there the last two years and we want to get there again.
"Some people maybe forget how hard it is to get there through the qualifiers; these teams are into their season and it's difficult. We need to do that before we start thinking ahead but it's where everyone wants to be and that's our aim."
As well as Elfsborg, Shakhter Karagandy would prove obdurate opponents for Celtic in last season's qualifying rounds. A 2-0 loss in Kazakhstan in the first leg left Neil Lennon's side up against it, only for a 3-0 home win to take them into the group phase. That, Mulgrew conceded, should serve as a warning about how hard it may be this time around. "That was a difficult game," he recalled. "That's the type of game you come up against [in the Champions League], a hostile atmosphere. Every team is scrapping to get there. It doesn't matter about ability when a team wants it and the prize at the end is so big and makes it very difficult."
Mulgrew and the players have not had much of a chance to get to know Deila, and he them, but first impressions have been good. "It's been great so far - well organised and positive. It's been a good start.It's still early days but it's going well.
"There are differences [from last season] but he's still trying to get his ideas across at the moment. I'm sure he's looked at games and seen certain things we did well but there will be things he'll think we can improve upon.
"I'm sure he'll get his ideas across and have his own way of thinking. Maybe some things will stay the same but there will be some ideas he will want to implement.
"We had a couple of meetings to get to know him and to get an idea of the way he wants to play and how he wants things to happen. It was really positive and everyone's responded well."