to come back, to do ... and come back
Verb て form +
When the verb
Action and return
This usage indicates that the speaker will go somewhere to perform an action and then return to the starting point.
牛乳を買って
I will go buy some milk (and come back).
郵便局へ行って
I'm going to the post office (and coming back).
Movement toward the speaker
This usage describes a movement or action that is physically directed toward the speaker's current location.
鳥が飛んで
A bird flew over here.
友達が家に遊びに
A friend came to visit my house.
When used as an auxiliary verb to show direction or returning,