Let k be a field of characteristic zero. Etingof and Kazhdan (Sel. Math. (N.S.) 2:1–41, 1996) construct a quantisation Uℏb of any Lie bialgebra b over k, which depends on the choice of an associator Φ. They prove moreover that this quantisation is functorial in b (Etingof and Kazhdan in Sel. Math. (N.S.) 4:213–231, 1998). Remarkably, the quantum group Uℏb is endowed with a Tannakian equivalence Fb from the braided tensor category of Drinfeld–Yetter modules over b, with deformed associativity constraints given by Φ, to that of Drinfeld–Yetter modules over Uℏb (Etingof and Kazhdan in Transform. Groups 13:527–539, 2008). In this paper, we prove that the equivalence Fb is functorial in b.