Add secondary time series to SARIMA model

Question

I have my primary time series which is monthly temperature. I have the following model for it.

    fit <- arima(Test, c(1,0,0), seasonal = list(order=c(0,1,1), period =12), include.mean = FALSE)

I also have monthly CO2 concentrations over the same time period, and I want to use this to see if I can improve my model, but I don't understand how to do the pre-whitening process. I have gotten the model and residuals for the CO2 time series, but help after this step is appreciated.

    fitCO2 <- arima(CO2, c(0,1,1), seasonal = list(order=c(0,1,1), period =12), include.mean = FALSE)
    CO2resd <- fitCO2$residuals
    fitCO2$coef
   

Also, how do you write out the expanded model for a SARIMA(1,0,0)(0,1,1)_12. I have seen examples but I'm not really following them.