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We address the inverse problem for holomorphic germs of a mapping of the complex line near a fixed point which is tangent to the identity. We provide a preferred parabolic map 
$\Delta $ realizing a given Birkhoff–Écalle–Voronin modulus 
$\psi $ and prove its uniqueness in the functional class we introduce. The germ is the time-
$1$ map of a Gevrey formal vector field admitting meromorphic sums on a pair of infinite sectors covering the Riemann sphere. For that reason, the analytic continuation of 
$\Delta $ is a multivalued map admitting finitely many branch points with finite monodromy. In particular, 
$\Delta $ is holomorphic and injective on an open slit sphere containing 
$0$ (the initial fixed point) and 
$\infty $, where the companion parabolic point is situated under the involution 
${-1}/{\mathrm {Id}}$. One finds that the Birkhoff–Écalle–Voronin modulus of the parabolic germ at 
$\infty $ is the inverse 
$\psi ^{\circ -1}$ of that at 
$0$.
