Grammar
-자고 하다
to report a suggestion or invitation
Form
Verb + -자고 (reported suggestion).
Function
-자고 reports a suggestion or invitation to do something together, often recalling what was said.
Why this form?
-다고 하다 — “to say that…”: reports a statement, not a suggestion to act together.
Examples
사이좋게 지내자고. — [She said] we should get along.
같이 가자고 했어요. — He suggested we go together.