\begin{definition}[Definition:Kernel (Category Theory)/Definition 1]
Let $\mathbf C$ be a category.
Let $A$ and $B$ be objects of $\mathbf C$.
Let $f : A \to B $ be a morphism in $\mathbf C$.
Let $\mathbf C$ have an initial object $0$.
A '''kernel''' of $f$ is a morphism $\map \ker f \to A$ which is a pullback of the unique morphism $0 \to B$ via $f$ to $A$.
\end{definition}