D^\infty$ as $\hat V$-marked curves. Then $C$ and $D$ are isomorphic.

### Step 5: Induction on $\dim V$ and the remainder of the proof {#step-5-induction-on-dim-v-and-the-remainder-of-the-proof .unnumbered}

We prove the Main Theorem in the $\dim V=1$ case. In the general case, the Key Lemma allows us to use induction on $\dim V$ to show that $C\cong f_C^*{{\cal C}}_{{\cal F}}$ as follows. There is a natural morphism $p\colon U_{{\cal F}}\to U_{{{\cal F}}'}$, and we show that the pullback $p^*{{\cal C}}_{{{\cal F}}'}$ is isomorphic to the $V'$-contraction $({{\cal C}}_{{\cal F}})'$ of ${{\cal C}}_{{\cal F}}$. We consider the $V'$-contraction $C'$ of $C$ and the corresponding morphism $f_{C'}$ constructed in the same manner as $f_C$. We show that $f_{C'}=p\circ f_C$. Using induction, we obtain a chain of isomorphisms $$C'\cong f_{C'}^*{{\cal C}}_{{{\cal F}}'}\cong (p\circ f_C)^*{{\cal C}}_{{{\cal F}}'}\cong f_C^*(p^*{{\cal C}}_{{{\cal F}}'})\cong f_C^*({{\cal C}}_{{\cal F}})'.$$ Using the fact that contraction commutes with pullback, the last scheme is isomorphic to the $V'$-contraction of $f_C^*{{\cal C}}_{{\cal F}}.$ We then show that the contractions to the $\infty$-components $C^\infty$ and $(f_C^*{{\cal C}}_{{\cal F}})^\infty$ are isomorphic. The may thus apply the Key Lemma to deduce that $C$ and $f_C^*{{\cal C}}_{{\cal F}}$ are isomorphic.

Finally, we show that the pullbacks of the universal family 