morphism of schemes induces a map of points

Let $f:X\to Y$ be a morphism of schemes over $S$, and let $T$ be a particular scheme over $S$. Then $f$ induces a natural function from the $S$-points of $X$ to the $S$-points of $T$.

Recall that a $T$-point of $X$ is a morphism $\varphi :T\to X$. So examine the following diagram:

Since all the schemes in question are $S$-schemes, the solid arrows all commute. The dashed arrow $\psi$ we simply construct as $f\circ \varphi$, making the whole diagram commute. The $\psi$ is a $T$-point of $Y$.