A metaphor that stuck with me when programming non-DI systems is "a person playing with his/her toys". A person being an object and the person's toys being anything that object creates, stores, initializes, and manipulates. The toys can dispatch events when they reach a certain state, but they know nothing about the person using them; they're just little black boxes with control switches that make up their interfaces. The person can listen for events from the toys and respond by manipulating their interfaces. The person can do whatever he/she wants with his/her toys, but he/she probably shouldn't go meddling with their innards because they might break.
The havoc that DI wreaks upon my metaphor is that it turns the toys into conscious beings that know their owner, the person using them. The toys can manipulate this person, but the person knows nothing of how they work and doesn't even care. The person just owns the toys and expects the toys to manipulate him/her to the toys' own satisfaction.
WTF?? That sounds horrible!!
What is a good mental metaphor they've been using to think of how DI systems work?