Explicit correspondence between classical double and quantum double

Proposition 12.3 of Etingof and Schiffmann's "Lectures on Quantum Groups" states the following claim.

Proposition 12.3. Let $H$ be a quantized enveloping algebra and let $\mathfrak{g}$ be thequasi-classical limit of $H$. Then $D(H)$ is a quantizationof $D\mathfrak{g}$.

Is there a specific way to write down the construction of this proposition when $\mathfrak{g}$ is a borel subalgebra of $\mathfrak{sl}(2, \mathbb{C})$?

I found a quantum double construction of $U_h(\mathfrak{b}_+)$ in Proposition 14 on p.258 of Klimyk and Schmiidgen's book "Quantum Groups and Their Representations".

The proposition gives the following commutation relation.

$$[H,\tilde{H}]=0, \quad[H,\tilde{F}]=-2\tilde{F}$$$$[E,\tilde{H}]=-2\hbar E, \quad[E,\tilde{F}]= \hbar\dfrac{e^{\hbar H}-e^{-\hbar \tilde{H}}} {e^{\hbar}-e^{-\hbar \tilde{H}}} $$

I think that if we set $\hbar$ in this proposition to 0, we do not get $D\mathfrak{b}_{+}$.