I reset my computer, now I have the Mac Os Yosemite Version 10.10.5 Now I can't update my system, due to two-factor authentication for Apple ID. The Yosemite version does not give me the small screen to put the code received on my phone.

I understand that. You put the password followed by the code into the same box at the same time. See the following from my previous reply:

Then type your password followed by the six-digit verification code directly into the password field.

See the following instructions from Two-factor authentication for Apple ID - Apple Support

If you use two-factor authentication with devices running older OS versions—like an Apple TV (2nd or 3rd generation)—you might be asked to add your six-digit verification code to the end of your password when signing in. Get your verification code from a trusted device running iOS 9 and later or OS X El Capitan and later, or have it sent to your trusted phone number. Then type your password followed by the six-digit verification code directly into the password field.