# Equivalency of the control function approach and IV

I am trying to see how these two are equivalent. So say Y = BX1 + e1, and x1 is endogenous. and say I have a Z s.t. E[Z'e]=0, and say the linear project of x1 onto z is given by X1= piZ + N. Now, if I were to use instrumental variable approach, I would get estimate my parameter by:

cov(piZ,y)/var(piZ), which I can rewrite as

cov(X1-N,y)/var(X1-N)

and if I use a control function approach estimating y = BX1 + pN + e2, my parameter would be estimated by (assuming x1 = qN + l)

cov(X1-qN,y)/var(X1-qN).

Now these arent these only equivalent if q =1?