No, he didn't do that properly. Otherwise, there'd be no need for three Kx constants.
The integrator term accumulates the error per sample by some law/filter and the Kd is multiplied by a filtered version of the difference between the error this sample and the error of the previous sample.
Actuator = (error * Kp) + ( filter1(sum(error)) * Ki ) + ( filter2(delta(error,lastError)) * Kd)
where the filters are very likely extremely stateful and not very filter-ey.
Forms exist for continuous linear PID and they're about what you'd expect.