Wadge-like degrees of Borel bqo-valued functions

We unite two well known generalisations of the Wadge theory. The first one considers more general reducing functions than continuous in classical case, and second extends reducibility from sets (i.e., { 0 , 1 stretchy="false">} \{0,1\} -valued functions) to alttext="upper Q"> Q encoding="application/x-tex">Q functions, for a better quasiorder . In this article, we consider reducibilities on generalise some results L. Motto Ros [J. Symbolic Logic 74 (2009), pp. 27–49] direction T. Kihara A. Montalbán [Trans. Amer. Math. Soc. 370 (2018), 9025–9044] direction: Our main result states that structure alttext="bold upper Delta Subscript alpha Superscript 0"> Δ α<!-- α </mml:msubsup> encoding="application/x-tex">\mathbf {\Delta }^0_\alpha -degrees plus gamma + γ<!-- γ }^0_{\alpha +\gamma }</mml:annotation> -measurable is isomorphic beta β<!-- β }^0_\beta</mml:annotation> }^0_{\beta these are generalized homomorphism order alttext="gamma"> encoding="application/x-tex">\gamma -th iterated -labeled forests.