Do nonfaithful functors appear in real colimit constructions?

[fallgesetz]

The colimit is defined in terms of a functor from an indexing category to another category(a diagram).

In the examples I've seen of the colimit, the functor is always faithful, so my question is, are there any nonfaithful functors from index categories that come up in real life?

 

[Hurkyl]

What about particular cases of the diagrams you already know?

e.g. the coproduct of an object with itself.