I am working on an exercise trying to show that $Spec \ k$ is final in category of $k$ schemes. I am stuck and I would appreciate any assistance. Thank you!
PS The definition I have for $k$ scheme is that it is a morphism of the form $X \rightarrow \operatorname{Spec} \ k$. And then I know from the exercise I did that $X \rightarrow \operatorname{Spec} \ A$ are in natural bijection with ring morphisms $A \rightarrow \Gamma (X, O_X)$.
So I figured if I have a $k$ scheme, then it follows that there exists a corresponding ring morphism $k \rightarrow \Gamma (X, O_X)$. I guess I was wondering how this is unique.