Kernels and cokernels are universal
For an arrow $f:E\to F$ in an abelian category $\mathcal{A}$, consider the category of "arrows into $E$ whose composites with $f$ are the zero arrow". An arrow in this category between two objects $X\to E$ and $X'\to E$ is an arrow $X\to X'$ in $\mathcal{A}$ forming a commutative triangle over $E$. A kernel $k:E'\to E$ is just a terminal object in this category, and hence is unique up to unique isomorphism by a standard argument.