Implication Details

Assumptions: additive, effective congruences, finitely complete

Conclusions: normal

Reason: Let $i : Y \hookrightarrow X$ be a monomorphism. Then the pullback $E$ of $X \times X \xrightarrow{p_1 - p_2} X \xhookleftarrow{~~~i~~~} Y$ is a congruence on $X$ . This is 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 $h$ .