Morphism of schemes determined by their induced maps of $Z$ valued points

Question

I am doing an exercise that states: morphism of schemes $X \rightarrow Y$ is determined by their induced maps of $Z$ valued points, as $Z$ varies over all schemes. I am a bit confused with this question (My attempted solution seems way too simple...) I would appreciate any assistance! Thank you!

Answer 1

For any morphism of schemes $\alpha: X \to Y$, and for any scheme $Z$, denote the induced map of $Z$-valued points $$\alpha^Z_*: X(Z) \to Y(Z)$$ which is defined as $\alpha^Z_*(\varphi)=\alpha \circ \varphi$.

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Let $\pi, \rho: X \to Y$ be two morphisms of schemes, and suppose for all schemes $Z$, we have $\pi^Z_*=\rho^Z_*: X(Z) \to Y(Z)$. Then, in particular, letting $Z=X$, we have $\pi^X_*: X(X) \to Y(X)$, $\rho^X_*: X(X) \to Y(X)$ satisfying $$\pi=\pi^X_*(\mathrm{Id}_X)=\rho^X_*(\mathrm{Id}_X)=\rho.$$