I am reading in Ralf Fröbergs notes on Koszul algebras. He writes that for any quadratic algebra $A$ there is a natural differential $$d:A_i\otimes (A^!_j)^*\to A_{i+1}\otimes (A^!_{j-1})^*$$ I think this differential $d$ can be defined as the composition, $$A_i\otimes (A^!_j)^*\to_{id\otimes d'}A_i\otimes(A_1^!\otimes A_{j-1}^!)^*\to A_i\otimes(A_1^!)^*\otimes (A_{j-1}^!)^*\to A_i\otimes A_1\otimes (A_{j-1}^!)^*\\ \to A_{i+1}\otimes (A^!_{j-1})^*$$ where $d'$ maps $f\in (A^!_j)^*$ to $d'f\in (A_1^!\otimes A_{j-1}^!)^*$ by defining $d'f(x\otimes y):=f(xy)$. The rest of the maps are natural.
Is this a correct understanding of the differential? If so, how do I show that $d^2=0$?
Much greatful for any response=)
