Implication Details

Claim: If a category is Cauchy complete and has kernel pairs and is preadditive, then it has kernels.

Proof: Let $f : X \to Y$ be a morphism. Since $f$ has a kernel pair, the functor $$\C^{\op} \to \Set^+, \quad T \mapsto \{(u,v) \in \Hom(T,X)^2 : f \circ u = f \circ v\}$$ is representable. Using $w = u-v$, this functor is isomorphic to $$T \mapsto \{(u,w) \in \Hom(T,X)^2 : f \circ w = 0\}.$$ The functor $T \mapsto \{w \in \Hom(T,X) : f \circ w = 0\}$ is a retract of this functor. Since the category is Cauchy complete, every retract of a representable functor is representable. Therefore, the kernel of $f$ exists.