Let $X, Y, Z$ be schemes, where $X, Y$ are $Z$ schemes. I know the definition of "the locally closed subscheme of $X$ where two $Z$- morphisms $\pi, \pi': X\rightarrow Y$ agree" from its universal property. Also I can define it as the fiber product of the diagonal $$\delta : Y\rightarrow Y\times_Z Y$$ with $$(\pi, \pi'): X\rightarrow Y\times_Z Y.$$
My question: how to prove that the underlying set of "the locally closed subscheme where the two morphisms agree" is the same as the set of points where the two morphism agree on the residue field.
It is probably clear thatthe former is contained in the latter, but why is it all of them? That is, why is a point where $\pi, \pi'$ agree on the residue field necessarily contained in "the subscheme where $\pi, \pi'$ agree"?