The association between polycystic ovarian syndrome (PCOS) and obesity is complicated. Signs and symptoms of PCOS begin for some females soon after they start having periods. Women with PCOS produce too much insulin, or the insulin they produce does not work as it should. The inability of insulin to function normally is one reason why women with PCOS tend to gain weight or have a hard time losing weight. For others, PCOS develops later on, following substantial weight gain. What is clear is that women affected by obesity have a greater risk for PCOS and women with PCOS have a greater risk for obesity.
I would like to refer you to this fabulous Bidirectional Mendelian randomization paper by our collogues that show "increased BMI [body mass index] is causal for PCOS while the reverse is not the case."
Bidirectional Mendelian randomization to explore the causal relationships between body mass index and polycystic ovary syndrome
Human Reproduction, Volume 34, Issue 1, January 2019, Pages 127–136, https://doi.org/10.1093/humrep/dey343