It's not as bad as you might imagine. They first administer a strong sedative. Over 10 minutes or so, your pet progressively gets calmer and calmer - their pains dissolve at last, and they lapse into a peaceful unconsciousness. In your lap, or you with it on the floor is it's a large dog. You caress it, pet it. Your eyes meet steadily for the last time. On their eyes is the look of relief...you will see that. Like people, they prefer to have their hands (paws) held at the end by loved ones... Finally, the heart-stopping injection is made.
Of course you will be weepy. As so you should, James.