Homological Algebra/Defintion of abelian category

Exercises

 * 1) Given $$f : a \to b$$ in an $$Ab$$-enriched category with zero object. Prove that $$f = 0_{a,b}$$ iff $$f$$ factors through $$\mathbf{0}$$.


 * 1) Given a biproduct $$(a \oplus b, i_1, p_1, i_2, p_2)$$ of $$a$$ and $$b$$. Prove that $$(a \oplus b, i_1, i_2)$$ is a coproduct of $$a$$ and $$b$$ and $$(a \oplus b, p_1, p_2)$$ is a product of $$a$$ and $$b$$.


 * 1) In an $$Ab$$-enriched category with zero object, a kernel of $$f: a \to b$$ can be equivalently be characterized as a pullback of $$\mathbf{0} \to b$$ along $$f$$.