The 24-year-old left the Parkhead club for Wolves four years ago after his fledgling career came to a halt under Strachan.
He returned north of the border to Aberdeen in 2008 and, after impressing current Celtic boss Neil Lennon, was snapped up by the Northern Irishman on a three-year deal as a free agent.
Mulgrew was joined yesterday by 29-year-old South Korea defender Cha Du-ri, who signed initially on a two year deal, subject to a work permit.
The former Celtic youth player knows there is likely to be stiff competition for places under Lennon's regime and is not taking his place next season for granted.
"Obviously it's a massive club and it will be hard to get into the team," he told Celtic's television channel, Channel 67.
"But it's up to me to work hard, improve over pre-season and show I'm good enough to play here.
"It's a great club and there are great players here so it's going to be hard.
"But I was down in England and I got a feel for what it was like down there which has made me a better player.
"I've had a couple of setbacks but I've always believed in myself, in my ability.
"I'm determined and I work hard.
"I don't like to look back to the past. It's all about the future now.
"A club the size of this will always be tough. You need to prove yourself every single day but I know that and I will look to do that."
Mulgrew denied there was a clash of personalities with Strachan but claimed personal development makes him better placed to take advantage of a rare second chance at Celtic.
He said: "At the time things weren't working here under that manager and it was time to move on.
"I wouldn't say it was a personal situation. The relationship was fine. It was for football reasons.
"Different managers have different opinions on players and maybe I wasn't his type of player at the time.
"Obviously you want to get chances coming through at a club and I was disappointed that I didn't get a proper chance but when I got the opportunity to move on I had to take it.
"I had to play games - that was the most important thing for me at that time.
"These things happen in football and you just have to move on and believe you can make a career for yourself and I was happy to do that at the time.
"It's up to me now. I think I have matured as a person.
"It's maybe taken a couple of setbacks for me to be the person I am now.
"But I feel that I am definitely a better player and a better person as well."