scale which is $f_n\circ M_{n,v}^{-1}$. This lemma insure that by looking at the right scale for the image we get a non constant limit.

Now, the magic of the right definitions works and it follows that if $j_n$ converges to a tree of spheres $\T^Z$, then the scales represented corresponding to the spheres of $\T^Z$ are exactly the ones given by Lemma \[postcomp\].

\[toscover\] Let $(f_n,i_n,j_n)_n$ be a sequence of rational maps marked by the same portrait ${\bf F}$. If $([i_n],[j_n])\to([{\cal T}^Y],[{\cal T}^Z])\in \overline{\rm Mod}_Y\times \overline{\rm Mod}_Z,$ then $(f_n,i_n,j_n)_n$ converges to a cover between trees of spheres $${\cal F}:{\cal T}^Y\to {\cal T}^Z.$$

Where covers between trees of spheres and convergence are defined below.

${\cal F}:{\cal T}^Y\to {\cal T}^Z$ is :

-   a trees map $F:T^Y\to T^Z$

-   for $w:=F(v)$, a ramified cover $f_v: \S_v\to \S_w$ s.t. 0.2cm

    1.  the restriction $f_v: \S_v-a_v(Y)\to \S_w-a_w(Z)$ is a cover; 0.2cm

    2.  adjacent attaching points maps to the corresponding adjacent attaching points 0.2cm

    3.  same local degree on both sides of edges.

Let ${ \F}:{\T}^Y\to { \T}^Z$ be a cover between trees of spheres of portrait ${\bf F}$. A sequence ${ \F}_n:=(f_n,a_n^Y,a_n^Z)$ of marked spheres covers converges to ${ \F}$ if their portrait is 