The Scot was booed by the travelling fans of his former club when the two sides met at Old Trafford in December but insists he is proud of what he achieved at Goodison Park
Moyes, who left the Merseyside club at the end of last season, was booed by the supporters who once adored him when the teams met in December.
The 50-year-old Scot played down the significance of his return to Everton on Sunday, but feels he deserves more respect from his former club's fans.
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"They had their own reasons for it [in December]," Moyes said.
"There were a few problems with us trying to sign a couple of their players, but overall I was just doing my job for Manchester United.
"We sourced a big majority of the players at Everton and put them towards the team. I think we did everything in our power at Everton to make sure we got the best team we could.
"This game isn't about me. It's about two teams, both of whom are still in need of winning the game.
"The focus of attention shouldn't be on me, it should be on the players, but I go there and I understand the Everton supporters have to get right behind their team."
Moyes bought midfielder Marouane Fellaini from Everton to Old Trafford for £27.5 million on the final day of the transfer window and has also targeted left-back Leighton Baines.
"Maybe people don't believe how it [signing with United] happened, but it happened two weeks before the end of the season and you'd have to say it couldn't have been done much better," he added.
"I think there was also animosity because of the players. We'd like to have bought a couple of them here.
"But lots of United players go to Everton - so there is a reverse.
"Hopefully in time everyone will look at it and say it was a really good period in Everton's history."