Van Nistelrooy, who acted as caretaker manager for the past fortnight, has left the club this week after the arrival of new manager Ruben Amorim.
Eriksen said from the Denmark camp today: "Of course, it's a shame as we wanted Ruud to stay as long as possible, but a new coaching staff is coming in.
"I think they would like some fresh eyes and their own people in, and I fully understand that.
"We can only thank Ruud for the time he was here. It was fun - even for the rest of the staff. It was a good time.
"Everyone knows Ruud Van Nistelrooy, and of course we have also seen him in the role of assistant coach before. But seeing him as a head coach has been fun.
"Of course it was temporary, and we were told that too, so it was a time of lots of joy and a lot of freedom that made the results fun enough to be reversed as well.
"As you have seen, he is still a giant, giant figure in United's history. Also for all the fans. He is the one they are yelling for."