My question is similar to the question in The image of the diagonal map in scheme. I saw a hint to my question in the comments, but I was not able to prove it.
Let $f:X\longrightarrow Y$ is a morphism of schemes. Let $T$ be a scheme and $h:T\longrightarrow X\times_Y X$ be a morphism such that $p_1\circ h=p_2\circ h$, where $p_1,p_2$ are projection morphisms from $X\times_Y X\longrightarrow X$. Then is it true that $h(T)\subset\Delta(X)$?
I initially tried proving that if $x\in X\times_Y X$ such that $p_1(x)=p_2(x)$, then $x\in\Delta(X)$. But that I think is not true, and I am going nowhere. I will be help if someone can tell me how to prove it.