In my experience:
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Cursor’s “Chat” needs you to explicitly give it access to files. I assume the codebase being indexing offers “Chat” the ability to show you a list of context (after you click the + button).¹
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Cursor’s “Composer” in “agent” mode (not “normal” mode) will implicitly have access to your entire project.
¹ This seems good for security.
² This seems bad for security. See Security Concerns with .gitignore, .cursorignore, .cursorban concern #2.