Compatibility between truncation and coproducts for quantum affine algebra and Yangian of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$

We prove that the standard Drinfeld-Jimbo coproducts for Yangians and quantum affine algebras factorize through their truncated quotients in the case of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$. As an auxiliary result, we give formulas for the coproduct of the truncation series in both cases.