Equivalence of Definitions of Kernel of Morphism

Theorem
Let $C$ be a category with zero object $0$.

Let $f : A \to B$ be a morphism in $C$.

Then the following definitions of kernel of $f$ are equivalent.

Definition 1 implies Definition 2
By Definition of Zero Object, $0$ is an initial object, so Definition 1 is possible.

Let $f : A \to B$ be a morphism in $C$.

Let $k : K \to A$ be a pullback of $f$ along the zero morphism $0 : 0 \to B$.

We check the universal property of the equalizer of $f$ and the zero morphism $0 : A \to B$.

Suppose $T$ is any object and $h : T \to A$ is any morphism with $f \circ h$ = $0 \circ h$.

By Composition with Zero Morphism is Zero Morphism $f \circ h = 0 : T \to B$.

By Definition it follows, that $h : T \to A$ and $0 : T \to 0$ is a cone on the pullback diagram defined by $f$ and $0 : 0 \to B$.

There is a unique morphism $t : T \to K$ with $0 \circ t = 0 : T \to 0$ and $k \circ t = h$.

It follows, that $k$ is an equalizer $\mathrm{Eq}\paren{f,0}$.