When we hand over a task or project to a colleague, the delegator will ask, 'Do you have the ball on that'?
Yes, I have the ball', means the colleague is now fully responsible.
For some of us, it's a small ceremony. (Any more formal and we'd bow.)
Fully responsible doesn't necessarily mean that the colleague is responsible for the whole task or will necessarily complete it, but that's all the more reason to be clear about what holding the ball entails.
You might say something like, 'Yes I have the ball on calling the client, then once she's happy with the arrangements, I'll hand it back to you to order the equipment for the day'. Wait for the nod or words of agreement.
It should always be clear who has the ball, or tasks can disappear into the ether and people say, 'Oh, I thought you were doing that!'.