He is talking about the progress Celtic have made from losing 3-0 away to SC Braga two years ago to defeating Barcelona at Celtic Park two weeks ago, but he might as well be discussing his own development. That night against Braga, Mulgrew was making his first start since rejoining Celtic, but his return to the club was initially a difficult experience. Now, Mulgrew is lauded and, even if he is used in a variety of positions, he tends to be an integral figure.
He had to readjust, and there were moments of uncertainty. They seem remarkable now, since Mulgrew often plays with aplomb and the supporters have even composed a chant in praise of him. Against Barcelona, Mulgrew played on the left of midfield, but dropped to full-back every time Adam Matthews stepped infield to strengthen the middle of Celtic's defence. The role demanded concentration and discipline from Mulgrew, but the reliability now comes naturally.
There was a time when he succumbed to a more haphazard character. His relationship with Gordon Strachan suffered due to Mulgrew's attitude as a young player at Celtic, and Billy Stark recalls moments of impetuosity during Mulgrew's Scotland Under-21 career, when he was sent off after receiving a second yellow card for taking off his shirt to celebrate a late equaliser against Finland, who then scored a winning goal.
Mulgrew also settled an argument with Ross McCormack about who should take a penalty kick against Slovenia with a game of paper, scissors, stone.
Yet last midweek against Luxembourg, Mulgrew played faultlessly for Scotland, controlling the midfield under Stark's interim management.
"I wasn't too chuffed with that penalty [against Slovenia]," said Stark. "We are always looking for personalities but we missed the penalty. [But] there was a real assurance in his play [against Luxembourg]. You felt he was in control along with Darren [Fletcher] and did dictate the play."
His Celtic manager, Neil Lennon, believes Mulgrew's best position is centre-back, but he is unlikely to play him there. Efetobore Ambrose and Kelvin Wilson will start, if fit, against Benfica in the Champions League on Tuesday, but Mulgrew will still be an influential presence. From midfield, he is better able to make use of his delivery of the ball, particularly from wide areas, and his passing range will dictate much of Celtic's ability to counter-attack.
This will be the first time that Celtic and Mulgrew return to Portugal since the Braga defeat, which was comprehensive. As a team, and as an individual, the game against Benfica represents an opportunity to emphasise how much distance has been travelled. Depending on the outcome of Spartak Moscow's game against Barcelona, a score draw in Portugal might be enough to guarantee Celtic's presence in the knockout stages.
"We've improved massively [since Braga]," Mulgrew said. "We're a lot more prepared for that type of challenge. The manager was just new and it was a new team trying to gel together, but he's been here for a couple of years now and you see that on the pitch. It takes a bit of time to settle at a club like Celtic and although I'd been at the club previously, it was all new faces. We've moved on since then. We've got a lot more belief in ourselves."
Qualifying from the group stages was considered unlikely for Lennon's team when the draw was made, but there has been a sophistication to his side in Europe this season. The 2-1 win over Barcelona was reward for tactical astuteness, but also players understanding their roles and the importance of following their instructions. For all that Victor Wanyama, Fraser Forster and Tony Watt were acclaimed, the performer who typified Celtic's approach was Kris Commons, since he suppressed his natural attacking instincts to play the same diligent, conscientious role as Mulgrew, on the opposite flank.
Mulgrew says the Celtic players always "had belief" that the team could finish in the top two, but the performance in both games against Barcelona has provided the substance to the squad's confidence. It also presents Celtic with an opportunity to seal their own fate without relying on other sides having to drop points, since a win in Portugal would be enough to put them through.
The game against Benfica will be taxing, since the Portuguese side will approach it as a fixture they must win, but Celtic's defensive stoicism, and ability to strike on the break, will be formidable.
"Because Benfica have to win, it will be our most difficult game in the group," Mulgrew said. "They'll throw everything at us, so we have to be prepared for that and rise to the challenge. It was a great result against Barcelona but we haven't done anything yet. We knew it would be tough, and we'd need a bit of luck and we've had that, but there's still work to be done. We can't just say because we beat Barcelona that we can take anyone on."
Celtic are on the verge of a significant achievement, but the team need to repeat the level of performance that brought them this opportunity.