Specialization of curves defined over function field

Question

Let $C$ and $D$ be two algebraic curves defined over a function field $K(t)$ ($K$ is a number field). Suppose that $C$ and $D$ are isomorphic. We can specialize the curves to values in $K$ (nonformally this can be done by just pluggin in a number in $K$ instead of $t$). This scenario is the same as reducing from a generic fiber to a special fiber. Is it true that the specialization of the isomorphism will remain an isomorphism for all but finitely many values in $K$?

I am not sure if this is a natural thing to do, or if this is the best way to present the question. I appreciate any help on this.

Answer 1

Writing out the isomorphism and its inverse map, all one has to do is to care that the denominators of the finitely many coefficients from $K(t)$ which appear in these two maps don't vanish under the specialization.