Proposition 4

Let S and $S'$ be tori. Denote by $M(S,S')$ the group of morphisms of Lie groups from S to $S'$. The maps $f \rightarrow X(f)$ and $f \rightarrow \Gamma (f)$ are isomorphisms of groups from $M(S,S')$ to Hom$_{\mathbf{Z}}(X(S'),X(S))$ and to Hom$_{\mathbf{Z}}(\Gamma (S), \Gamma (S'))$, respectively.

If $f$ is a morphism of Lie groups from S to $S'$, the homomorphism $X(f)$ is simply the dual of $f$ in the sense of Spectral Theories, Chap. II, §1, no. 7. The map $\varphi \rightarrow \widehat{\varphi}$ from Hom$_{\mathbf{Z}}(X(S'),X(S))$ to $M(S,S')$ defined in loc. cit. is the inverse of the map $f \rightarrow X(f)$ from $M(S,S')$ to Hom$_{\mathbf{Z}}(X(S'),X(S))$; the latter is thus bijective. If we identify $\Gamma (S)$ (resp. $\Gamma (S')$) with the dual $\mathbf{Z}$-module of X(S) (resp. $X(S')$) (Prop. $3$)$,\Gamma (f)$ coincides with the transpose of the homomorphism $X(f)$, hence the proposition.