  In this paper, we show the equivalency between the existence of fast neutrino
flavor instability and that of neutrino flavor lepton number (NFLN) crossings.
The veracity of this proposition has been uncertain and sometimes controversial
despite it being essential in the flavor evolutions of dense neutrinos. This
study clarifies the conditions under which fast instability occurs and
contributes to the elucidation of collective neutrino oscillations.
