Let $\mathfrak{ g}$ be a symmetrizable Kac-Moody algebra with associated Yangian $Y_\hbar\mathfrak{ g}$ and Yangian double $\mathrm{ D}Y_\hbar\mathfrak{ g}$. An elementary result of fundamental importance to the theory of Yangians is that, for each $c\in \mathbb{ C}$, there is an automorphism $\tau_c$ of $Y_\hbar\mathfrak{ g}$ corresponding to the translation $t\mapsto t+c$ of the complex plane. Replacing $c$ by a formal parameter $z$ yields the so-called formal shift homomorphism $\tau_z$ from $Y_\hbar\mathfrak{ g}$ to the polynomial algebra $Y_\hbar\mathfrak{ g}[z]$. We prove that $\tau_z$ uniquely extends to an algebra homomorphism $\Phi_z$ from the Yangian double $\mathrm{ D}Y_\hbar\mathfrak{ g}$ into the $\hbar$-adic closure of the algebra of Laurent series in $z^{ -1}$ with coefficients in the Yangian $Y_\hbar\mathfrak{ g}$. This induces, via evaluation at any point $c\in \mathbb{ C}^\times$, a homomorphism from $\mathrm{ D}Y_\hbar\mathfrak{ g}$ into the completion of the Yangian with respect to its grading. We show that each such homomorphism gives rise to an isomorphism between completions of $\mathrm{ D}Y_\hbar\mathfrak{ g}$ and $Y_\hbar\mathfrak{ g}$ and, as a corollary, we find that the Yangian $Y_\hbar\mathfrak{ g}$ can be realized as a degeneration of the Yangian double $\mathrm{ D}Y_\hbar\mathfrak{ g}$. Using these results, we obtain a Poincar\'{ e}-Birkhoff-Witt theorem for $\mathrm{ D}Y_\hbar\mathfrak{ g}$ applicable when $\mathfrak{ g}$ is of finite type or of simply-laced affine type.
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