Celtic secured one of the club's most famous victories when they beat Barcelona in the Champions League earlier this month. But some of the supporters who hailed the players as heroes that night booed them off the park following Caley Thistle's 1-0 victory at Celtic Park on Saturday.
"That's what happens at a club like Celtic," said Mulgrew, whose team have a swift chance to bounce back when they face Hearts at Tynecastle on Wednesday night.
"You get a good result and it is quickly forgotten about and on to the next game. That is the way it has to be.
"There is a lot of expectation, especially at home in the league.
"We have to live up to that and win the games and take whatever stick the fans give us. We need to take responsibility.
"On Saturday, we didn't take a couple of chances and that can lead to a defeat.
"We are all focused on getting a result in the next match and we take full responsibility for a couple of bad results."
Celtic have taken just one point from a possible nine in the SPL at Parkhead and manager Neil Lennon felt the wrath of some fans in angry exchanges during and after the match. Lennon soon said he would "do the honourable thing" and go if fans were not happy with him. But Mulgrew believes the remonstrations and Lennon's subsequent threat only show how much the club means to him and fans. The Scotland international said: "I think it is heat of the moment. Maybe they caught him straight after the game or whatever, with tempers running high it is something you say.
"But I don't think the manager will be leaving, he is right behind the cause here."