[v5 ます-stem] + ましょうか [v1 ます-stem] + ましょうか [する → し] + ましょうか [来る → き] + ましょうか
The form ましょうか is used in two main ways: offering to do something for someone ("Shall I…?") and suggesting doing something together ("Shall we…?"). It is formed by adding か to the volitional polite form ましょう.
The most common use of ましょうか is to offer assistance to the listener. The speaker is volunteering to do something on behalf of or for the benefit of another person.
In this usage, the subject is always "I" (the speaker). The listener can accept or decline the offer.
ましょうか can also be used to suggest doing something together with the listener. Context makes it clear whether the speaker means "Shall I…?" or "Shall we…?"
When a question word like なん何、どこ、or いつ is included, it almost always means "Shall we…?" rather than "Shall I…?"
Both ましょうか and ませんか can be used to make suggestions, but they have different nuances. ませんか is an invitation that gives the listener room to decline ("Won't you…?" / "Would you like to…?"), while ましょうか is more of a collaborative suggestion that assumes both parties will participate ("Shall we…?").