A generic nonlinear optimal control problem with a Bolza cost functional is discretized by a Legendre pseudospectral method. According to the covector mapping theorem, the Karush-Kuhn-Tucker multipliers of the discrete problem map linearly to the spectrally discretized covectors of the Bolza problem. Using this result, it is shown that the nonlinear programming problem converges to the continuous Bolza problem at a spectral rate assuming regularity of appropriate functions.