A man wearing a straw hat sitting in a chair holding the leash on a dog who is laying on the sidewalk next to him.
A man is holding the leash of a dog.
A man holding the leash 'on' a dog is a man holding the leash 'of' a dog.
The leash of a dog implies the leash is on a dog.
Holding the leash of a dog and holding the leash on a dog can be considered the same thing.