Doesn’t dependency injection work against lazy initialisation?

I am trying to design my java classes, and for making them unit testable I am trying to incorporate dependency injection as much as possible.
But doing so also makes me think that making client code to inject collaborators would mean that we may need to further push it back to client’s client and so on. So we may end up creating the collaborator long before it’s really needed. Isn’t it against lazy initialisation?