2013 Toyota Corolla 1.8 manual swap

I have a 2013 Toyota corolla 1.85 speed and I would like to know if I can buy a transmission six speed from a 2014 Corolla will it fit Or can this be done
I will start by saying I am no expert, however, the engine is the same between the two. The transmission changed from a 5 speed to a 6 speed. Will the six speed bolt up to the 2013 engine? I can't see why it would not, however there is likely software that would need to be updated or changed so the 2013 knows it is dealing with a six speed. I have asked this question to myself several times, but I don't really want to go to the trouble of researching to find out what it takes.