A covariant functor $F: \mathcal{A} \to \mathcal{B}$ between abelian categories is:

• Left exact if it preserves finite limits and the exactness of sequences $0 \to A \to B \to C$
• Right exact if it preserves finite colimits and the exactness of sequences $A \to B \to C \to 0$
• Exact if it preserves both left and right exactness

Given functors $F, G: \mathcal{C} \to \mathcal{D}$, a natural transformation $\eta: F \Rightarrow G$ assigns to each object $X \in \mathcal{C}$ a morphism $\eta_X: F(X) \to G(X)$ such that for any morphism $f: X \to Y$:

$G(f) \circ \eta_X = \eta_Y \circ F(f)$

This commutativity is called naturality.

A category $\mathcal{C}$ is additive if:

1. Each $\text{Hom}(A, B)$ is an abelian group
2. Composition is bilinear
3. Finite products and coproducts exist and coincide (direct sums)
4. There exists a zero object

An additive category is abelian if additionally:

• Every morphism has a kernel and cokernel
• Every monomorphism is a kernel and every epimorphism is a cokernel

In an abelian category $\mathcal{C}$:

• An object $P$ is projective if $\text{Hom}(P, -)$ is exact
• An object $I$ is injective if $\text{Hom}(-, I)$ is exact

Equivalently, $P$ is projective if every epimorphism $E \twoheadrightarrow P$ splits, and $I$ is injective if every monomorphism $I \hookrightarrow E$ splits.