A natural extension of the Wadge hierarchy is obtained if we consider k-partitions (or even Q-partitions for a countable better quasiorder Q) of the Baire space instead of sets (i.e., 2-partitions). Natural initial segments of the Q-Wadge hierarchy were recently characterised in terms of the so called h-quasiorder on suitably iterated Q-labeled countable well founded forests. Though these characterisations are natural and constructive, their definitions are rather long and technical. In this paper, we find clear shorter characterizations as (reducts of) a kind of free structures satisfying some natural axioms. Informally, we provide short and clear axiomatisations of the initial segments.