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.
For any morphism of $S$-schemes $f:X\to Y$ the graph morphism $\Gamma_f:X\to X\times _SY$ is an immersion (but not necessarily a closed immersion).
Its associated image subscheme $G_f\subset X\times _SY$ is indeed isomorphic to $X$ under the restriction $g:G_f\to X$ of the first projection $p_X:X\times _SY \to X$.
This is explained in Corollaire (5.3.11) page 133 of $EGA_I$ and the seven following lines .