Let L be a commutative ring, M a free L-module of finite rank m ⩾1, and u an endomorphism of M. There exists a unique endomorphism u~ of M satisfying the relation
u~(x)∧w=x∧∧m−1(u)(w)
for x∈M and w∈∧m−1(M). Moreover, we have the relations
u∘u~=u~∘u=det(u)M,
det(u~)=det(u)m−1,
u~=r=0∑m−1(−1)rcm−1−r(u)ur.
Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.