Multicoloured Clique is NP-hard: every language in NP reduces to it by a map computable in polynomial time by a Turing machine [FHRV09], [CFK+15, Theorem 13.7].

Independent Set is NP-hard, as proved in the archive ($Lax762056$). The reduction of $WH_F4_IndependentSetToMcc$ is computable in polynomial time on the word RAM, hence by a Turing machine ($Lax759944$), and preserves the answer. Composing a reduction to Independent Set, the rewriting of its output bits as a word of numbers, and this reduction gives a reduction to Multicoloured Clique. The reduction to Independent Set always produces an instance, so the composite always produces the word of a Multicoloured Clique instance ($mcc_npHard_in$), which is what a further reduction that is correct only on instances needs.