Kagawa was unable to exhibit his best form at Old Trafford following a move from Dortmund, where he won two Bundesliga titles, in 2012.
Having struggled to nail down a regular starting berth in his preferred playmaker role, the 26-year-old was deemed surplus to requirements by new United boss Louis van Gaal and sealed a return to Signal Iduna Park on Sunday.
Writing in his latest blog, Mata said of his former team-mate: "I would like to wish all the best to Shinji Kagawa, a fantastic player but especially a great person.
"It's been a pleasure to be his team-mate and I'm sure both Borussia Dortmund and fans everywhere will enjoy his football again."
Mata also commented on the new faces at United, hailing the capture of Angel di Maria from Real Madrid and the prospective arrival of Netherlands international Daley Blind from Ajax.
"I would like to welcome Di Maria and Blind to the club," added the Spaniard.
"I am sure these two players will give a lot to us and will strengthen the team.
"We already know what Angel is capable of, his ability and dribbling skills, and Daley has a great touch and is very intelligent, as he showed during the World Cup."
