  This is the third in a sequence of three papers in which we prove the
following generalization of Thomassen's 5-choosability theorem: Let $G$ be a
finite graph embedded on a surface of genus $g$. Then $G$ can be $L$-colored,
where $L$ is a list-assignment for $G$ in which every vertex has a 5-list
except for a collection of pairwise far-apart components, each precolored with
an ordinary 2-coloring, as long as the face-width of $G$ is at least
$2^{\Omega(g)}$ and the precolored components are of distance at least
$2^{\Omega(g)}$ apart. This provides an affirmative answer to a generalized
version of a conjecture of Thomassen and also generalizes a result from 2017 of
Dvo\v{r}\'ak, Lidick\'y, Mohar, and Postle about distant precolored vertices.
In a previous paper, we proved that the above result holds for a restricted
class of embeddings which have no separating cycles of length three or four. In
this paper, we use this special case to prove that the result holds in the
general case.
