Pick up after your dog.

The photo credit goes to M. The other day she sent me this from the US, where she's been for a week.

It seemed to me that the phrase, 'pick up after your dog,' was not perfect as it lacked either an object (what you are expected to pick up) or some words to follow 'your dog' (the reference as to what your dog has done). However I've learned that it's 'pick up after' that works as it is, and I've found some other examples using the phrase, such as 'You should pick up after yourself' and 'I'm always picking up after the boy.'