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Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with associatedYangian $Y_\hbar\mathfrak{g}$ and Yangian double$\mathrm{D}Y_\hbar\mathfrak{g}$. An elementary result of fundamental importanceto the theory of Yangians is that, for each $c\in \mathbb{C}$, there is anautomorphism $\tau_c$ 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 $\hbar$-adicclosure of the algebra of Laurent series in $z^{-1}$ with coefficients in theYangian $Y_\hbar\mathfrak{g}$. This induces, via evaluation at any point $c\in\mathbb{C}^\times$, a homomorphism from $\mathrm{D}Y_\hbar\mathfrak{g}$ intothe completion of the Yangian with respect to its grading. We show that eachsuch 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 adegeneration of the Yangian double $\mathrm{D}Y_\hbar\mathfrak{g}$. Using theseresults, we obtain a Poincar\'{e}-Birkhoff-Witt theorem for$\mathrm{D}Y_\hbar\mathfrak{g}$ applicable when $\mathfrak{g}$ is of finitetype or of simply-laced affine type.

The formal shift operator on the Yangian double