Proposition 9

If $B$ is flat over $A$, homomorphism (13) is bijective. Indeed $\psi'(L(E) \otimes_A B, B \otimes_A L(F))$ is bijective (X, p. 102, prop. 1) and $\gamma$ is bijective (X, p. 66, cor. 2).

Suppose $B$ flat over $A$. Substituting $E$ for $Q$, $B$ for $N$ and $B \otimes_A F$ for $M$ in homomorphism (12), we obtain a homomorphism

which is bijective by Proposition 8. Substituting $E$ for $M$, $F$ for $N$, $A$ for $B$, $B$ for $Q$, in homomorphism (10), and interchanging the factors in the tensor products, we obtain a homomorphism of $B$-modules

whence by composition a homomorphism called canonical

(14)