For every linear mapping u of an A-module E into an A-module F, the linear mapping y* ↦ y* ∘ u of the dual F* of F into the dual E* of E is called the transpose of u and is denoted by t u.
The transpose tu is therefore defined by the relation (15)
⟨u(x),y∗⟩=⟨x,tu(y∗)⟩for all x∈E and all y∗∈F∗.
Definition 5 applies without alteration to right A-modules and is then equivalent to the relation
⟨y∗,u(x)⟩=⟨tu(y∗),x⟩for all x∈E and all y∗∈F∗.
Formulae (9) and (10) of § 1, no. 2 here give (16)
t(u1+u2)=tu1+tu2
for two elements u1,u2 of HomA(E,F) and (17)
t(v∘u)=tu∘tv
for u∈HomA(E,F) and v∈HomA(F,G), G being a third A-module; finally, clearly (18)
t1E=1E∗.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.