Date: 202108
Local time of day: 0601-1200
Location: ZZZ.TRACON, US
Flight conditions: VMC
Weather elements / visibility: 20
Light: Daylight
Aircraft: Citation V/Ultra/Encore (C560)
Operator: Air Carrier
Operating under: Part 135
Flight plan: IFR
Mission: Passenger
Flight phase: Climb
Airspace: Class C ZZZ
ATC / advisory: TRACON ZZZ
Aircraft component: Turbine Engine
Component problem: Improperly Operated; Malfunctioning
Reporter organization: Air Carrier
Reporter function: First Officer; Pilot Flying
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Flight Instructor; Flight Crew Instrument; Flight Crew Multiengine
Human factors: Communication Breakdown; Situational Awareness; Time Pressure; Troubleshooting
Anomaly: Aircraft Equipment Problem Critical; Deviation / Discrepancy - Procedural FAR; Deviation / Discrepancy - Procedural Maintenance; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Person Flight Crew
When detected: In-flight
Result: Aircraft Aircraft Damaged; Flight Crew Took Evasive Action; General Flight Cancelled / Delayed; General Maintenance Action
Contributing factors: Aircraft; Human Factors; Procedure
Primary problem: Human Factors

Narrative: I was pilot flying; from the left seat; and the company identified PIC was pilot monitoring from the right seat. On the departure procedure from ZZZ; the right engine engine-fire light illuminated. I reduced the right throttle; as per the memory items for the indication; and the light extinguished immediately.We continued on the departure procedure and noted that the oil temperature for the right engine was at red-line. I asked the PIC/pilot monitoring to review the checklist for procedures after the memory items related to the engine fire light; while I monitored the engine gauges for other indications if a problem; radios and autopilot tracking of the departure procedure. The aircraft continued to climb within the parameters of the SID with reduced thrust on the right engine.PIC/pilot monitoring stated that we were good to continue and proceeded to look in the emergency procedures for something related to high oil temperature. PIC/pilot monitoring stated that there were no procedures or checklists for high oil temperature and that we should continue the flight at a reduced power setting on the right engine. The power setting for the right engine was about 88% N2; while the left engine was about 94% N2.PIC/pilot monitoring stated that as long as we could maintain an oil temperature below red line and climb per the SID and at a rate of at least 500 feet per minute; we should continue the departure. He did not state we should consider returning to land the aircraft.We completed the flight as he directed and called maintenance for assistance; after landing. On removing the engine cowling; significant heat damage was noted; probably from a bleed air leak; according the maintenance provider on site. Also; a metal conduit from the front to the back of the engine was found to have not been attached at the rear connecting point.Both engines were changed a week or two before this event; about 10 flight hours; ago.After the flight; I had a chance to review the procedures and found that a landing as soon as practical is called for in the procedure. This was not mentioned by the PIC.I discussed this with the PIC later in the day; and he said that he had not seen that information during the flight. Retrospectively; I am not convinced that he actually read the procedure beyond the memory items and was distracted by the high oil temperature indication and possibly by outside factors such as other work and the desire to complete the flight. PIC stated he hoped the repairs could be completed quickly so we could continue our scheduled trips in the coming days. I am also not convinced the PIC understands the gravity of his actions; or lack thereof; during this flight.There are several issues related to this event.The failure to properly install the engine was most likely the cause of the engine fire indicator light and the high oil temperature indication. I am baffled by the fact that the issue did not occur sooner after the engine change.The failure of the PIC to correctly read and follow the procedures in the emergency check list and for him to state that the flight should continue could have caused catastrophic outcome to the event.My failure; as pilot flying; to insist on an expeditious landing regardless of the checklist and the PIC's decisions; after the engine fire warning light; also decreased the chances of a successful outcome of the flight. I considered declaring an emergency and landing; but did not execute that procedure.




Synopsis: Citation V First Officer reported that during a Fire Warning event in flight; the Captain declined to follow checklists and established procedures and elected to continue the flight to destination.