Lemma 1

Suppose that the A-module F is projective. The group homomorphism $\theta$ is injective, and its image is Hom$^f_A(E,F)$.

By loc. cit., Corollary, the homomorphism $\theta$ is injective. Its image is contained in Hom$^f_A(E,F)$. Let $u\in$ Hom$^f_A(E,F)$; let us prove that $u$ belongs to the image of $\theta$.

First, suppose that the A-module F is free. Let $(f_i)_{i\in I}$ be a basis of F, and let $(f_i^*)_{i\in I}$ be the basis of $F^*$ dual to $(f_i)_{i\in I}$. Let J be a finite subset of I such that the image of $u$ is contained in the submodule of F generated by the $f_j$ for $j\in J$. We have

for every $x\in E$, and therefore

In the general case, there exist a free A-module L and homomorphisms $i: F\rightarrow L$ and $p: L\rightarrow F$ such that $p\circ i= 1_F$. The homomorphism $i\circ u$ belongs to Hom$^f_A(E,L)$. By the above, there exist a finite set J and, for every $j\in J$, elements $x^*_j$ of $E^*$ and $y_j$ of L such that we have

for every $x\in E$. We then have

for every $x\in E$, and therefore $u=\theta (\sum_{j\in J}x^*_j\otimes p(y_j))$.