Implication Details

Claim: If a category is additive and has effective congruences, then it is normal.

Proof: Let $i : Y \hookrightarrow X$ be a monomorphism. Then we define a relation on $X$ via $E \coloneqq X \times Y$ with maps $f, g : E \rightrightarrows X$ defined by $f : (x, y) \mapsto x+i(y)$ and $g : (x, y) \mapsto x$. It is straightforward to check that $f$ and $g$ are jointly monomorphic. Now $E$ is a congruence because for generalized elements $x_1, x_2 \in X(T)$, $(x_1, x_2)$ factors through $E$ if and only if $x_1 - x_2$ factors through $Y$. In other words, the relation on $X(T)$ is exactly $x_1 \equiv x_2 \pmod{Y(T)}$, which is an equivalence relation on $X(T)$ (and in fact a congruence in $\Ab$). Now by assumption, $E$ is the kernel pair of some morphism $h : X \to Z$; in other words, $(x_1, x_2)$ factors through $E$ if and only if $h(x_1) = h(x_2)$. In particular, for $x \in X(T)$, $x$ factors through $Y$ if and only if $(x, 0)$ factors through $E$, which is equivalent to $h(x) = h(0) = 0$. We have thus shown that $Y$ is the kernel of $h$.