Despite being satisfied with Celtic's recent accomplishments on the domestic scene, Lennon is now looking for the Glasgow giants to step up in the UEFA Champions League.
"Now we are stamping our authority on the game in Scotland and want to take it further," he told BBC Scotland.
"We want to take it into Europe. There's room for improvement obviously - there are other competitions we want to look at and strive forward in.
"But it is in our DNA as a club to win championships and, since I've been involved in the club, that's nine in 14 years.
"It was our priority and we wanted to improve on last year. This is unquestionably the best league campaign I've had as a manager.
"We are the title favourites every season, but we've been emphatic this year."
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