Building on what we learned about ましょう, you'll recall that it's a polite form derived from the auxiliary verb ます. It's used to propose a shared activity, similar to 'let's' in English.

By attaching the question particle か, the nuance shifts from a simple suggestion ('let's') to a more direct question ('shall we?'). This makes the expression more inquisitive than using ましょう alone.

Similar to ましょう, this grammar pattern is formed by attaching it to the stem of a verb (the pre-ます form).

• カフェでコーヒーを飲(の)みましょうか。
  Shall we have some coffee at the cafe?
• 週末(しゅうまつ)に映画(えいが)を見(み)ましょうか。
  Shall we see a movie this weekend?

It's important to understand that these examples function as 'suggestions', not 'invitations'. Japanese, much like English, distinguishes between these two concepts. To extend a clear invitation to someone, it is more natural to use ませんか, a form closely related to ましょうか.

• 一緒(いっしょ)に図書館(としょかん)へ行(い)きませんか？
  Won't you go to the library with me?
• 一緒(いっしょ)に図書館(としょかん)へ行(い)きましょうか。
  Shall we go to the library together?

The main distinction is that a 'Won't you...?' question (ませんか) feels like a more personal invitation, while a 'Shall we...?' question (ましょうか) is more of a mutual suggestion for a joint activity.