In the paper "On the consistency strength of projective uniformization" Woodin proves a lemma "Assume $M$ is a model of ZFC that is $\Sigma^{1}_{3}$-absolute. Then $M\vDash\forall x\in\mathbb{R}\,[x^{\sharp} \mathrm{\ exists}]$." He then goes on to say after the proof "It in fact now follows by a theorem of Martin-Solovay [6] that $\Sigma^{1}_{3}$-absoluteness is equivalent to the existence of $S^{\sharp}$ for every set $S$."

[6] Martin, D. A. and Solovay, R. M., A basis theorem for $\Sigma^{1}_{3}$ sets of reals, Ann. of Math. 89 (1969), 138-160.

When I read this article of Martin and Solovay I have trouble seeing the connection with the assertion that $S^{\sharp}$ exists for all sets $S$. I was wondering if anyone could clarify this for me.