Moyes instilled a great amount of defensive qualities into Everton’s team during his time in charge, while Martinez enhanced the attacking quality in the side.
However, neither manager quite managed to get the balance entirely correct, and Carragher feels that the next boss to come in at Goodison Park must ensure that Everton are strong both defensively and going forward.
“I think his [Martinez's] first season was outstanding,” Carragher said to Sky Sports. “It was a mix of his ideas and what David Moyes had left him. And that would be the perfect recipe for Everton to try and get into that top four.
“They had a mix of defensive steel with the attacking style that Martinez likes.”
Martinez was sacked by Everton last week, ending his three year period in charge of the Toffees.
The Merseyside club have badly unperformed this term, and they have ended up finishing in 11th place in the Premier League table.
Martinez failed to address the defensive issues in Everton’s side, with the problems seemingly becoming more severe rather than better.
Everton are now on the look out for a new manager, with Ronald Koeman and Frank De Boer amongst the favourites to land the job.