Is the topological dual of a quantized enveloping algebra a Hopf algebra?

I am looking for a proof of Proposition 10.2 in Etingof and Schiffmann’s Lectures on Quantum Groups. In particular, I would like to understand the proof of the following statement in an elementary way:

Let $H$ be a quantized enveloping algebra. Then the dual Hopf algebra $H'=\mathrm{Hom}_K(H,K)$ is an $\hbar$-formal group.

Here, $K$ denotes $k[[\hbar]]$, where $k$ is a field of characteristic zero.

A similar claim appears in Chari–Pressley’s A Guide to Quantum Groups on p.190, right after Definition 6.3.3, as well as in Lemma 2.1 of Gavarini’s paper "The quantum duality principle". However, I have not been able to find an explicit proof in these references.

Could someone provide a detailed proof of this result, or point me to a reference where a proof is given? I would especially appreciate an approach that is as elementary as possible.