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This is the third article in the series begun with [BonWon3, BonWon4],devoted to finite-dimensional representations of the Kauffman bracket skeinalgebra of an oriented surface $S$. In [BonWon3] we associated a classicalshadow to an irreducible representation $\rho$ of the skein algebra, which is acharacter $r_\rho \in \mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$ represented bya group homomorphism $\pi_1(S) \to \mathrm{SL}_2(\mathbb C)$. The main resultof 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 shadowof an irreducible representation of the Kauffman bracket skein algebra. We alsoprove that the construction used in our proof is natural, and associates toeach group homomorphism $r\colon \pi_1(S) \to \mathrm{SL}_2(\mathbb C)$ arepresentation of the skein algebra $\mathcal S^A(S)$ that is uniquelydetermined up to isomorphism.

Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality