Changing the units changes the scale and shape of Weibull pdf?

I've the following question:

I've determine the Weibull parameters using the ML method for a given input data in meters. Changing the units to mm changes the scale and shape of Weibull pdf? I know the shape is unitless and the scale has units equal to the variable, so if it doesn't affect it means that the scale now is 1E3 times the initial scale and the shape remains the same.

Thanks