Unifiers modulo commutativity in terms of syntactic unifiers and $\approx_{C}$-class
Note that for all $s, t \in T(\Sigma, V)$ we have:
$$s \approx_{C} t \iff \exists_{i, j}:  s_i = t_j \tag{*}$$
Using (*) for the terms $s \sigma$ and $t \sigma$ we obtain:
$$s \sigma \approx_{C} t \sigma \iff \exists_{i, j}: s_i \sigma = t_j \sigma, $$i.e,
$$\mathcal{U}_{C}(\{ s \approx_{C}^{?} t \}) = \bigcup\limits_{1 \leq i \leq m \\ 1 \leq j \leq n} \mathcal{U}_{\emptyset}(\{s_i =^{?} t_j \})$$