Let $S$ be a scheme and let $f:X\to Y$ be a morphism of separated $S$-schemes. This MSE answer shows that, when $X$ is reduced, the obvious morphism $g_f:\Gamma_f \to X\times_S Y\to X$ is an isomorphism, where $\Gamma_f$ is the scheme-theoretic image of $\Delta_f:X\to X\times_s Y$.
Can we show that $g_f$ without assuming that $X$ is reduced?
I see how the reduced induced structure is being used there, but I do not see if $X$ being reduced affects the statament, $X$ cannot have double/triple/... points if $\Gamma_f$ does not! So, I wonder if there is a probably more elaborate proof that shows $g_f$ to be an isomorphism.
The only assumption I have is that $S$ is Noetherian.
A hint or a reference would be appreciated.