What is the kernel of a pair of morphisms in $\mathbf{Set}$? Suppose we're in the category $\mathbf{Set}$. Then the kernel of a pair of morphisms $f,g\colon A\to B$ is a third morphisms $h\colon C\to A$ such that for morphism $k\colon D\to B$ such that $fk=gk$, there is a unique $j\colon D\to C$ such that $hj=k$. 
What exactly is $h\colon C \to A$ in this category? I'm curious what the set $C$ and function $h$ would be, and then I think I could verify that it actually satisfies the desired properties.

$h$ is essentially the inclusion of $C=\{x\in A:f(x)=g(x)\}$; i.e. exactly the set of inputs on which $f$ and $g$ agree.
