  This is the third article in the series begun with [BonWon3, BonWon4],
devoted to finite-dimensional representations of the Kauffman bracket skein
algebra of an oriented surface $S$. In [BonWon3] we associated a classical
shadow to an irreducible representation $\rho$ of the skein algebra, which is a
character $r_\rho \in \mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$ represented by
a group homomorphism $\pi_1(S) \to \mathrm{SL}_2(\mathbb C)$. The main result
of the current article is that, when the surface $S$ is closed, every character
$r\in \mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$ occurs as the classical shadow
of an irreducible representation of the Kauffman bracket skein algebra. We also
prove that the construction used in our proof is natural, and associates to
each group homomorphism $r\colon \pi_1(S) \to \mathrm{SL}_2(\mathbb C)$ a
representation of the skein algebra $\mathcal S^A(S)$ that is uniquely
determined up to isomorphism.
