Limits and $Hom(-,Y)$-functor in abelian categories 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.
 A: 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 need 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.
