The most basic knowledge in the United States is the separation of powers, that is, the separation of judicial, legislative and executive powers.
Although the FBI and the CIA are both departments of the United States, the FBI belongs to the judicial branch, while the CIA is an intelligence agency and, to a certain extent, belongs to the president. So there is no connection between the two departments.
And the CIA's investigation is limited to investigation and cannot make arrests. Although the FBI has the authority, major incidents naturally require a response from the Department of Justice. CIA operations, unlike those of the FBI, are strictly secret. Even to a limited extent, Congress cannot intervene. Overall, the CIA is almost a presidential agency. Because of the separation of powers, one works for the president and the other works for the Department of Justice.
If the FBI cooperates with the CIA, it may be a serious threat to the security of the United States. For example, the 911 incident is serious enough to directly threaten the security of the United States. The FBI and CIA are still cooperating on this matter. But usually the two organizations perform their own duties and there is no overlap.