I understood that the Sabre was really a new design but based on the Fury the team had developed just before but that Phantom II and Corsair II were really adaptation. Adaptations in the other direction, from Air Force to Navy, didn't happen in the US in the jet era?
Well, the F-84 Sabre was purely a US Air Force design, and it came first, before the FJ-2 Fury. At the time, the US Navy was waiting for the F-7U Cutlass and the F-9F Cougar, but they were some time away from being ready for production, so as an interim measure the Navy decided to create a US Navy version of the F-84 and it was ultimately designated the FJ-2 Fury. Upgrades to that design were called the FJ-3 Fury.
The F-4 Phantom II was a US Navy design that entered service for the US Navy in 1960. The US Air Force liked what they saw and they adopted it, not requiring as heavy a landing carriage or arrestor hook. In USAF service the F-4 was initially called the F-110 Spectre. But in 1962, the US Military adoped the United States Tri-Service aircraft designation system, which changed the designation back to the F-4. The USAF F-4 design ultimately became the largest user of the Phantom II, with the F-4C, D, and E versions for Air Force use. The first full production aircraft entered service in 1964 for the US Air Force.
An interesting note about the F-4 Phantom II is that the US and Germany ultimately sent a total of 192 aircraft to Turkey who still operates many of them today, in 2014. One of the Turkish Recon F-4 Phantom IIs was the aircraft that was shot dwon in the Syrian conflict last year.
The A-7 Corsair II was a US Navy attack aircraft design with very good manueverability dog-fighting capabilites, and able to carry a lot of weight and loiter for long periods over the battle field. It was introduced into US Navy service in early 1967. It was another design that the US Air Force liked and adopted, the US Air Force version, the A-7D entered US Air Force service in 1970.
The US Air Force A-7D differed from the Navy's Corsair II in several ways. The Air Force wanted significantly more power for A-7D and selected the Allison TF41-A-1 turbofan engine, which was a license-built version of the Rolls-Royce Spey. It offered a thrust of 14,500 pounds, which was over 2000 pounds more than the TF30 engine that the Navy's Corsair IIs used. Other changes included a heads-up display, a new avionics package, a M61 rotary cannon in place of the two single-barreled 20-mm cannon, and a computerized navigation/weapons delivery system with AN/APQ-126 radar. In turn, the Navy liked and adopted the US Air Force improvements and the follow-on US Navy aircraft orders, which also had the strenghtened undercarriage, the arrestor hook, and the refueling probe, became the A-7E aircraft that were the main stay A-7 Corsaitrs for the US Navy.