what is the correct notation at the end of letter if you are writing it for someone?
On behalf of...........................

Acting on behalf of...................

Or sign yourself and put your relationship to the person you are writing for - Personal Assistant to x.

It does depend how formal the letter is who it is going to and in what capacity you are writing but the top 2 are pretty safe.
If you have transcribed a letter for someone, you can inform the reader this way:

At the bottom of the letter, left-aligned, type the initials of the person you are writing the letter for in upper-case, followed by a colon and your initials in lower-case.

Example:

Sally Wilson transcribed a letter from John Smith

Sincerely,

John Smith

CEO, Smith Inc.

JS: sw