Wikipedia says "A monomorphism is normal if it is the kernel of some morphism". But a kernel is an object and a monomorphism is a morphism. How can a monomorphism also be a kernel? Does it mean that its domain is the kernel of some morphism?
nLab's definition has the same issue: It says "A monomorphism $f:A\rightarrow B$ is normal if it is the kernel of some morphism $g:B\rightarrow C$ ". Does it meach that $f$ is normal if $A$ is the kernel of $G$ and $f$ is the morphism satisfying the properties that make $A$ the kernel?
A kernel is not an object. A kernel (likewise an equalizer) is a pair $(A,f)$ of an object $A$ and a morphism $f$, satisfying some universal properties (see nlab or Wikipedia articles about kernels). Actually, such pairs $(A,f)$ are determined by their morphisms $f$, so some authers prefer to define kernel simply as the corresponding morphism. To avoid ambiguity, sometimes the object $A$ of a kernel is denoted by $\text{Ker}(g)$ and the morphism of kernel by $\text{ker}(g)$.