Established in 1899 in Japan, Fuji is one of the great Japanese bicycle brands. I believe that Fuji was originally based in Ueno, not far from Osaka, but moved in the 1970’s to Machida, north east of Tokyo.
Highlights of its history include the exploits of Shoichiro Sugihara who won a gold medal at the first Asian Games in 1951 riding a Fuji, then became Fuji’s chief engineer and designed the bikes used by the Japanese National team at the totemic 1964 Olympic Games in Tokyo (and at the subsequent Mexico and Munich Olympics). Dr. Sugihara was determined that Fuji should produce real sports bicycles and rise above the great mass of utilitarian bicycles then produced in Japan.
Fuji was one of the Japanese brands that broke through into the US market in the early 1970’s establishing Japanese bicycle technology as the standard for others to meet. I certainly remember late 1970’s Fuji’s having a fantastic standard of equipment and quality of paint work - all at an alarmingly competitive price.
In terms of derailleurs Fuji did not, of course, manufacture or design anything themselves. They did, however, have the prestige and buying power to demand that derailleurs should be branded with their name.
At some point the Fuji brand was bought by Ideal, a major Taiwanese manufacturer which claims to be the third largest bicycle manufacturer in the world. The Fuji brand still exists and is designed and marketed from the US and manufactured by Ideal in Taiwan (and possibly in China and Poland as well).
There is some history on the Fuji web site.