When one has abelian categories, one is usually interested in additive functors. By definition, these are functors $F : \mathcal{C} \to \mathcal{D}$ whose action on morphisms is an abelian group homomorphism $\mathcal{C}(A, B) \to \mathcal{D}(F A, F B)$.
Proposition. If $\mathcal{C}$ and $\mathcal{D}$ are additive categories (i.e. $\textbf{Ab}$-enriched categories with finite direct sums) and $F : \mathcal{C} \to \mathcal{D}$ is an ordinary functor, then the following are equivalent:
- $F$ preserves finite coproducts (including the initial object)
- $F$ preserves finite products (including the terminal object)
- $F$ preserves the zero object and binary direct sums
- $F$ is additive
Proof. (1), (2), and (3) are equivalent because coproducts, products, and direct sums all coincide in an abelian category. One shows that (4) implies (3) by observing that being a direct sum in an $\textbf{Ab}$-enriched category is a purely equational condition: given objects $A$ and $B$, $(A \oplus B, \iota_1, \iota_2, \pi_1, \pi_2)$ is a direct sum of $A$ and $B$ if and only if
\begin{align}
\pi_1 \circ \iota_1 & = \textrm{id} &
\pi_1 \circ \iota_2 & = 0 \\
\pi_2 \circ \iota_1 & = 0 &
\pi_2 \circ \iota_2 & = \textrm{id}
\end{align}
$$\iota_1 \circ \pi_1 + \iota_2 \circ \pi_2 = \textrm{id}$$
where $\iota_1 : A \to A \oplus B$ and $\iota_2 : B \to A \oplus B$ are the coproduct insertions and $\pi_1 : A \oplus B \to A$ and $\pi_2 : A \oplus B \to B$ are the product projections.
On the other hand, (3) implies (4) by the following trick: given $f, g : A \to B$ in an abelian category $\mathcal{C}$, we have
$$f + g = \nabla_B \circ (f \oplus g) \circ \Delta_A$$
where $\Delta_A : A \to A \oplus A$ is the diagonal map and $\nabla_B : B \oplus B \to B$ is the fold map; this can be verified by using the last equation in the above paragraph:
\begin{align}
\textrm{id} \circ (f \oplus g) \circ \Delta_A
& = \textrm{id} \circ \langle f, g \rangle \\
& = (\iota_1 \circ \pi_1 + \iota_2 \circ \pi_2) \circ \langle f, g \rangle \\
& = \iota_1 \circ f + \iota_2 \circ g
\end{align}
and so $\nabla_B \circ (f \oplus g) \circ \Delta_A = \nabla_B \circ (\iota_1 \circ f + \iota_2 \circ g) = f + g$. Hence, if $F$ preserves the zero object and direct sums, it must also preserve addition of morphisms. ◼
One often also considers left/right exact functors between abelian categories. Officially, these are functors that preserve all finite limits/colimits (resp.), but in the case of abelian categories, it is enough that they be additive and preserve all kernels/cokernels (resp.). An exact functor is one that is both left and right exact.
These are all non-trivial conditions: the subject of homological algebra is essentially the study of the difference between left/right exact functors and exact functors! For example, $\textrm{Hom}(A, -)$ and $\textrm{Hom}(-, B)$ are both left exact functors; $\textrm{Hom}(A, -)$ is exact if and only if $A$ is a projective object, and $\textrm{Hom}(-, B)$ is exact if and only if $B$ is an injective object.
Limits always commute with limits. Here is a precise statement. Let $J_1, J_2$ be two diagram categories and let $F : J_1 \times J_2 \to C$ be a diagram in a category $C$. Then whenever the limits
$$\lim_{j_1 \in J_1} \lim_{j_2 \in J_2} F(j_1, j_2)$$
and
$$\lim_{j_2 \in J_2} \lim_{j_1 \in J_1} F(j_1, j_2)$$
exist, they are canonically isomorphic because they are both canonically isomorphic to the limit $\lim_{(j_1, j_2) \in J_1 \times J_2} F(j_1, j_2)$ of $F$ itself. For a discussion see Section 2.12 in Borceux. (To be totally precise, those inner limits are taking place in the category of functors $J_1 \to C$ resp. the category of functors $J_2 \to C$, but limits in functor categories are always computed pointwise so I think the abuse of notation is forgivable. You can think of this as a statement about adjoints composing.)
The description of when limits commute with colimits is more complicated. Perhaps the most important case is that filtered colimits always commute with finite limits in $\text{Set}$, and hence in any category $C$ equipped with a faithful functor $U : C \to \text{Set}$ preserving and reflecting filtered colimits and finite limits; in particular, categories of modules have this property. For a discussion see Section 2.13 in Borceux.
The corresponding statement for arbitrary abelian categories is false; in particular, there's no reason for it to be true in $\text{Ab}^{op}$. To write down a counterexample in $\text{Ab}^{op}$ means to find an example where cofiltered limits fail to commute with pushouts in $\text{Ab}$; as a special case this means to find an example where cofiltered limits fail to commute with cokernels.
Here is an explicit counterexample. Consider the cofiltered diagram of morphisms $p^n \mathbb{Z} \to \mathbb{Z}$ where $p$ is some prime. The cofiltered limit of the cokernels is the $p$-adic integers $\mathbb{Z}_p$, but the cokernel of the cofiltered limit is $\mathbb{Z}$ (the cofiltered limit itself is $\mathbb{Z} \xrightarrow{0} \mathbb{Z}$).
This phenomenon implies in particular that the cofiltered limit fails to be exact in general, and in some situations this means one has to take its derived functor, often denoted $\lim^1$.
Best Answer
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.