# Category of Abelian Groups: Limits

Let the category of Abelian Groups. I know that product and coproduct of a finite number of objects are the same in this category. Then, it follows that the projective and injective limits of finite diagrams are the same? Thanks.

• You expect to be able to generalize, from limits and colimits of finite diagrams with no arrows being equal, that limits and colimits are equal for any finite diagram? – Arthur Jan 21 '19 at 17:03

No. For instance, for a diagram of the form $$A\to B$$ with two objects and an arrow between them, the limit will be $$A$$ and the colimit will be $$B$$ (this is true in any category).