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We prove that if $F$ is a degree $3$ Thurston map with two fixed criticalpoints, then any irreducible obstruction for $F$ contains a Levy cycle. As acorollary, it will be shown that if $f$ and $g$ are two postcritically finitecubic polynomials each having a fixed critical point, then any obstruction tothe mating $f \mate g$ contains a Levy cycle. We end with an appendix to showexamples of the obstructions described in the paper.

Matings of Cubic polynomials with a fixed critical point, Part I: Thurston Obstructions