City boss Manuel Pellegrini brought Caballero to the club from Malaga for around £4m earlier this summer, and the England international has said that he is more than ready to fight for his place at the Etihad.
He told the Manchester Evening News: "It suits me. I'm a fighter, I've been a fighter all my life. I'll continue to do that. I'll carry on trying to perform to my best and hopefully keeping clean sheets.
"I wouldn't say it's given me an extra edge because I've always had an edge. Caballero is a great signing.
"But I've got the place at the moment and I'm going to do everything I can to keep it. You always want to look to improve - whether it's small margins or big margins."
Hart was selected in goal as City began their Premier League defence with a 2-0 win against Newcastle United on Sunday.