Add the required amount of shampoo to the toilet water and raise both the seat and the lid.
Get the cat and soothe it while you carry him/her to the bathroom.
In one smooth, swift movement, put the cat in the toilet and drop both the seat and the lid (you may need to stand on the lid to hold it down). CAUTION: Do not get any part of you body too close the edge as the cat's claws will be reaching out for any purchase they can find.
Flush the toilet three or four times. This provides a "power wash and rinse" that I find quite effective.
Have someone open the door to the outside and ensure there are no people between the toilet and the outside door.
Stand behind the toilet as far as you can and quickly lift both the seat and the lid.
The now clean cat will rocket out of the toilet and run outside where it will dry itself.
