A reduction of the NF consistency problem

We give a necessary and sufficient condition in order that a typeshifting automorphism be constructed on a model of the Theory of Simple Types (TST) by forcing. Namely it is proved that, if for every n ≥ 1 there is a model of TST in the ground model M of ZFC that contains an n-extendible coherent pair, then there is a generic extension M [G] of M that contains a model of TST with a type-shifting automorphism, and hence M [G] contains a model of NF. The converse holds trivially. It is also proved that there exist models of TST containing 1-extendible coherent pairs.