Lyth played seven Tests under Cook, who stepped down as England captain this week, during a testing period with England.
"He's an England legend," said Lyth.
"There were plenty of rumours circulating about Cooky stepping down, so it doesn't come as a massive surprise.
"It was a great honour to open the batting with him and play for him, and I can't speak highly enough of him.
"You could talk to him about anything and he is really passionate about England cricket.
"He'll go down as one of the best England captains, certainly of the modern era, and I'm sure he'll be looking forward to not having to go to all the press conferences and just thinking about sticking his pads on and going out to bat.
"Maybe this will be a weight off his shoulders, I'm sure he'll go on to score loads more runs for England and break many records."
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