 "A" a Lie algebroid (with anchor formula_24 and bracket formula_25), also its dual formula_23 a Lie algebroid (inducing the differential formula_27 on formula_28) and formula_29 (where on the RHS you extend the "A"-bracket to formula_30 using graded Leibniz rule).This notion is symmetric in "A" and formula_23 (see Roytenberg).Here formula_32 with anchor formula_33 and the bracket is the skew-symmetrization of the above in "X" and "α" (equivalently in "Y" and "β")