Let $f:Y\to Z$ be an etale morphism, $g_1, g_2:X\to Y$ be morphisms. Assume that for all closed points $x\in X$ we have $g_1(x)=g_2(x)$ and that $f\circ g_1=f\circ g_2$. Can I conclude (under some reasonable conditions on $X$, $Y$, and $Z$) that $g_1=g_2$?
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