Why does the map induce an injection of function fields

Question

I was reading the book Arithemtic of Elliptic curves by Silverman and came across this

Let C1/K and C2/K be curves and let $\phi$ : C1 → C2 be a nonconstant rational map defined over K. Then composition with $\phi$ induces an injection of function fields fixing K,

$$\phi^{*}:K(C_{2})\longrightarrow K(C_{1}),\ \ \ \ \ \phi^{*}f=f\circ\phi$$ Why is this an injection?