"If the tournament was being played just after the European Cup final, maybe it would have been fair enough for people to say we weren't particularly the best team to represent Europe," said Carragher.
"I don't think anyone could say that now. There aren't many teams in Europe who can be playing as well as we are. We're in the tournament on merit for our achievements last season and we're a much better side now than when we won the Champions League.
"We deserve to be here. I think we'll do ourselves, the Premiership and the rest of Europe justice.
"The manager said we would improve a few months ago and now we're proving it. Our results show that.
"Maybe we're starting to get a bit more respect for our performances and our achievements.
"We certainly deserve it for what we've done in the last eight months. If we can get the title champions of the world, we should get even more.
"There was a lot of discussion about whether or not we would enter after we had just won the European Cup. At one stage it didn't look as though we were going to come here, but I was always hoping we would.
"It means a lot to me to play in this competition. You're not going to get many opportunities in your career to play in the World Club Championship. It's a competition a lot of players don't even get the chance to take part in, never mind win. I'm sure it's something I'll be able to look back on in years to come and I would love to get a winners' medal.
"There's no question we're taking it seriously. We'll be disappointed if we don't win it. As far as I'm concerned, every trophy you can win matters, whether it's the Champions League or the Super Cup.
"The way I look at it is we're two wins away from another important trophy, and one which we may never get the chance to win again."