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.
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.