In Cartier (Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents. Sém. BOURBAKI, 53ème 2000–2001, no. 885), Pierre Cartier conjectured that for any non-commutative formal power series $\mathrm{\Phi }$ on $X=\{x_0,x_1\}$ with coefficients in a $\mathbb{Q}$-extension, $A$, subjected to some suitable conditions, there exists a unique algebra homomorphism $\phi$ from the $\mathbb{Q}$-algebra generated by the convergent polyzetas to $A$ such that $\mathrm{\Phi }$ is computed from the ${\mathrm{\Phi }}_{KZ}$ Drinfel’d associator by applying $\phi$ to each coefficient. We prove that $\phi$ exists and that it is a free Lie exponential map over $X$. Moreover, we give a complete description of the kernel of $\zeta$ and draw some consequences about the arithmetical nature of the Euler constant and about an algebraic structure of the polyzetas.