Example 9.2.5.2. The terminology of Definition 9.2.5.1 originates in homological algebra. Suppose that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are abelian categories (Definition 
) and that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an additive functor (that is, a functor which preserves direct sums). In this case, the following conditions are equivalent:
- $(1)$
The functor $F$ is left exact: that is, it preserves finite limits.
- $(2)$
The functor $F$ preserves equalizers.
- $(3)$
The functor $F$ preserves kernels. That is, for every exact sequence
\[ 0 \rightarrow X \xrightarrow {u} Y \xrightarrow {v} Z \]in the abelian category $\operatorname{\mathcal{C}}$, the image
\[ 0 \rightarrow F(X) \xrightarrow {F(u)} F(Y) \xrightarrow { F(v) } F(Z) \]is an exact sequence in the abelian category $\operatorname{\mathcal{D}}$.
The equivalence $(1) \Leftrightarrow (2)$ follows from Corollary 7.6.4.25 (since $F$ is assumed to be additive). The equivalence $(2) \Leftrightarrow (3)$ follows from the observation that, for every pair of morphisms $f_0,f_1: Y \rightrightarrows Z$, the coequalizer $\operatorname{Eq}(f_0, f_1)$ can be identified with the kernel of the morphism $(f_1 - f_0): Y \rightarrow Z$ (see Proposition ).