What's "an F/O's sector or leg?" Not such a simple question.
The question of “First Officers’ Sectors” or "giving legs" is actually more complex than is often realised, because there are so many different variations on the theme. These pages discuss it in the context of an airline that permits its Captains to allow "role reversal" in order to familiarise both pilots with the other's role, which is a very common situation.
On such sectors the F/O does the takeoff and landing, and in general all the in-flight tasks that would otherwise be done by the Captain. Typical legislation covering First Officer sectors define them as "Pilot-in-command under supervision (PICUS): A co-pilot performing, under the supervision of the pilot-in-command (PIC), the duties and functions of a pilot-in-command".
In its fullest form, this means that "Provided that the method of supervision is acceptable to the authority, a co-pilot may log as PIC flight time flown as PICUS, when all the duties and functions of PIC on that flight were carried out, in such a way that the intervention of the PIC in the interest of safety was not required."
The Captain performs the "Pilot Monitoring" tasks, while retaining overall responsibility for the flight as aircraft commander. The phrase "in-flight tasks" used here recognises that some practical restrictions may exist: for example, taxying may not be possible if nose wheel steering is only available from gthe Captain's (left hand) seat.
When laying down policies for PICUS flying, an operator will probably add other elements of supervision than simply safety. It will not usually want F/Os to make decisions with negative commercial consequences, either. It will not expect a flight operating as PICUS to be diverted, for example, if the Captain would not have made the same decision. It will also be expecting the aircraft and operating conditions (e.g. weather for the landing) will stay the pretty much the same as when the the Pilot in Command (i.e. the Captain) made his or her original decision to allow the First Officer to act as Pilot-in-Command Under Supervision.
So if a significant non-normal situation, or a major change in weather conditions occurs during such a flight, one of the first decisions which will need to be made is whether this new situation requires the initial pre-departure decision to allow a PICUS sector to be reversed.
Take an example flight. Prior to departure the destination weather forecast was good, and the Captain allocated the leg to the F/O. However on arrival, the RVR has dropped to 300m. This requires a Cat 2 approach, which under company rules requires the Captain to make the landing. It would hardly be good commercial sense to say "I've given the F/O this sector, he can't make the landing so we have to go to our alternate, even though we could land if it wasn't a PICUS sector".
So at this point it's usually expected that the Captain must resume the P1 role. During a "First Officer's sector" it is always the Captain’s responsibility to determine whether the circumstances have changed to the extent that it requires him to resume the Pilot-in-charge (P1) role, or whether he should continue to allow the First Officer to act as P1.
Consequently there is no discussion of non-normal or emergency operations on this site. It's assumed in those cases, either the Commander is the Pilot-in-charge, or that the Commander is happy to let the First Officer continue as Pilot-in-Charge under his or her supervision.