Remark 2

By Prop. 4, the map $f \rightarrow \Gamma (f)(2\pi i)$ from $M(\mathbf{U},S)$ to $\Gamma (S)$ is an isomorphism; if $a\in X(S) = M(S,\mathbf{U})$ and $f\in M(\mathbf{U},S)$, then the composite $a\circ f\in M(\mathbf{U},\mathbf{U})$ is the endomorphism $u \rightarrow u^r$, where $r=\langle a, \Gamma (f)(2\pi i)\rangle$. We shall identify $M(\mathbf{U},\mathbf{U}) = X(\mathbf{U})$ with $\mathbf{Z}$ from now on, the element $r$ of $\mathbf{Z}$ being associated to the endomorphism $u \rightarrow u^r$; thus, with the notations above,