On the coequaliser of a kernel pair How does one prove the following statement about kernel pairs?

If a pair of parallel morphisms is a kernel pair and has a coequalizer, then it is the coequalizer of its kernel pair.


I can't make any sense of this statement; in particular I don't understand what "it" refers to.
Here is the correct statement: suppose $f : X \to Y$ is a morphism with a kernel pair $g_1, g_2 : X \times_Y X \to X$ which is also the coequalizer of some other pair of maps $h_1, h_2 : Z \to X$. Then in fact $f$ must be the coequalizer of its kernel pair. For a proof, see this blog post. 
