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Let $\cal C$ be an abelian category and let $(\{x_i\} , \{\phi_{ij}:x_i\rightarrow y_j\}_{i\leq j})$ be a direct system with direct limit $(\varinjlim x_i, \phi_i: x_i \rightarrow \varinjlim x_i)$. Then the following holds: $$\varprojlim Hom(x_i,y) \cong Hom(\varinjlim x_i , y) $$

I know that if $\cal C$ is the category of $R$-modules the statement is true. Is it true for every abelian category?

My question came out while trying to prove that a left adjoint functor (between abelian categories) preserve direct limits through Yoneda Lemma.

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Yes, it is true in any abelian category. In fact, moreover, it's true in every category full stop that $$\hom(\text{colim}_i x_i,y)\cong \lim_i\hom(x_i,y).$$

(in category theory, a projective limit is often just called a limit, and an injective limit is often just called a colimit).

It's not only true when $x_i$ is a direct system (the category theoretic analogue of a directed poset is a filtered category). Indeed it's true for any shape diagram in $C$.

So the contravariant hom-functor takes colimits to limits. A dual result says that the covariant hom-functor takes limits to limits (one says it is a continuous functor),

$$\hom(y,\lim_i x_i)\cong \lim_i\hom(y,x_i).$$

The proof is not hard. By definition of colimit, an arrow out of $\text{colim}_i x_i$ is a cocone over the system $\{x_i\}_i,$ i.e. a commuting set of arrows out of the $x_i$.

Note also that while you don't your system to be directed in order to conclude the colimit commutes with $\hom(-,Y)$, directedness/filteredness is still relevant; filtered colimits commute with finite limits, at least for functors in some nice categories. This useful property is relevant to some exactness statements in homological algebra.

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  • $\begingroup$ Let me also point out that due to the Freyd-Mitchell embedding theorem, we know that every abelian category admits an exact embedding into a module category. Assuming the colimit in the module category is the same as in the abelian category (some details here are not clear to me), knowledge of this fact in the module category is sufficient to infer it for all abelian categories. $\endgroup$ – ziggurism Dec 19 '17 at 21:53
  • $\begingroup$ Note that filtered colimits commuting with finite limits is not enjoyed by every category; an example of an abelian category that doesn't have that property is $\mathbf{Ab}^{\mathrm{op}}$. $\endgroup$ – Hurkyl Dec 20 '17 at 5:45
  • $\begingroup$ @Hurkyl really? I thought it was not even particular to abelian categories, but in any category filtered colimits commute with finite limits. So is it only for Set-enriched categories and some Ab-enriched? $\endgroup$ – ziggurism Dec 20 '17 at 5:50
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    $\begingroup$ @ziggurism I thought that the Mitchell-Freyd Embedding Theorem only held for small Abelian categories? I mean if you're only ever working with a finite diagram you're fine, but I thought that you ran into problems with the embedding with certain infinite diagrams in large Abelian categories (and if I recall correctly I think that you can get into trouble doing this if you're working with $\mathcal{O}_X-{\mathbf{Mod}}$, the category of scheme modules for a nonaffine scheme $X$) . $\endgroup$ – Geoff Dec 20 '17 at 5:53
  • $\begingroup$ @ziggurism: It's an easy mistake to make. I make it a lot! A reference for this failure is the warning below proposition 3.3 at nlab (unfortunately, the link doesn't have much to say on the topic). An example of a Set-enriched category without this property is $\mathbf{Set}^{\mathrm{op}}$. $\endgroup$ – Hurkyl Dec 20 '17 at 5:54

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