Composition with Zero Morphism is Zero Morphism

Theorem
Let $\mathbf C$ be a category.

Let $0$ be a zero object in $\mathbf C$.

Let $A, B, C, D$ be objects in $\mathbf C$.

Let $f : A \to B$, $g : B \to C$ and $h : C \to D$ be morphisms in $\mathbf C$.

Let $g$ be a zero morphism.

Then $g \circ f$ and $h \circ g$ are zero morphisms.

Proof
By definition of zero morphism, $g$ is the composition of the unique morphism $\alpha : B \to 0$ with the unique morphism $\beta : 0 \to C$.

Since $\alpha \circ f : A \to 0$ is a morphism with codomain $0$ and $0$ is a terminal object, $\alpha \circ f$ is the unique morphism $A \to 0$.

It follows, that $g \circ f = \beta \circ \alpha \circ f$ is a zero morphism.

Since $h \circ \beta : 0 \to D$ is a morphism with domain $0$ and $0$ is an initial object, $h \circ \beta$ is the unique morphism $0 \to D$.

It follows, that $h \circ g = h \circ \beta \circ \alpha$ is a zero morphism.