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 $\mathrm{𝟎}$.

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 $\mathrm{𝟎}\to b$ along $f$.