Is terminal object the kernel of identity morphism?

Question

Let's say that there is a category $\mathbf{C}$ with $A$ being an object of that category and a zero object exists in that category. If we have an identity morphism ${id}_A: A\to A$, is the kernel of this morphism a terminal object of the category? My reasoning for this stems from the fact that the universal property of kernel requires a unique morphism going into $\ker(id)$, hence the terminal object.

Answer 1

EDIT: As pointed out in the comments by @Geoffrey Trang, the same argument applies to monomorphisms and dually to epimorphisms.

In fact, you can show that any isomorphism $f : A \longrightarrow B$ satisfies $ker(f) = 0$. Indeed, let $g: C \longrightarrow A$ such that $f \circ g = 0$. We want to show that $g$ factors through $0$. Well since $f$ is an isomorphism, $g = 0$. Hence, $g$ factors as $C \longrightarrow 0 \longrightarrow A$ and we have shown that $0$ satisfies the universal propertt of the kernel. Dually, the cokernel of an isomorphism is $0$.