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Let k be a field of characteristic zero. Etingof and Kazhdan constructed aquantisation U_h(b) of any Lie bialgebra b over k, which depends on the choiceof an associator Phi. They prove moreover that this quantisation is functorialin b. Remarkably, the quantum group U_h(b) is endowed with a Tannakianequivalence F_b from the braided tensor category of Drinfeld-Yetter modulesover b, with deformed associativity constraints given by Phi, to that ofDrinfeld-Yetter modules over U_h(b). In this paper, we prove that theequivalence F_b is functorial in b.

A 2-categorical extension of Etingof-Kazhdan quantisation