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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.

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  • $\begingroup$ 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? $\endgroup$ – Arthur Jan 21 '19 at 17:03
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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).

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