SNECMA's engines are all named M plus a number.
They never finished development of their M56 engine project, which was mostly intended for the Dassault Mercure airliner.
Due to delays with M56 development the Mercure went to the air with P&W JT8D power, and it consequently never met the originally intended range capability.
They did develop a good fan unit, but went into great troubles with the M56 core. Then they joined forces with GE
who had had Uncle Sam pay for development of the GE
F-101 engine for the B-1 bomber.
Basically they joined the core of the F-101 with the fan of the SNECMA M56 engine project.
terms I think that CF means "Commercial Fan engine", but I'm not sure. Anyway it's a fact that commercial fan engines from GE
are named CF-x.
So basically they assembled the name CFM56 the same way as they put the engine together.