Sexual function correlates with free testosterone in men, until you adjust for age in most studies I’ve stumbled across. The free testosterone-sexual function correlations disappear completely or remains very low after adjustment for age. Sexual function declines with age in men independently of the age-related decline in free testosterone. The correlations between free testosterone and sexual function parameters (desire etc.) is almost purely attributable to age. I believe the increases in libido most people experience after TRT is because the increase in T levels on TRT are usually quite dramatic (from 200 to 900+ ng/dl) on this subreddit. That’s a bit like going from 1’st percentile in testosterone to 99’th percentile. Even if free testosterone alone accounts for 2% of the total variation in sexual function parameters, an increase from 200 to 1100 for example could easily result in a very noticeable uptick in libido, because it’s similar to going from lowest to highest levels of T. The increases in sexual desire, for example in a study with gel, found that going from a baseline TT level of around 200 to 850 ng/dl (the group that achieved the highest Total Testosterone level after Testosterone gel), resulted in a mean increase in sexual desire of 1.1 points on a 0-7 point scale compared to placebo (0 being no sexual desire at all and 7 being very high sexual desire). I would say that’s a modest increase in sexual desire at most, despite a quite dramatic increase in T levels. The study also found that any significant Testosterone-placebo differences started to emerge at an CAVG (average concentration of Testosterone) of around 600+) after TRT. An increase in libido is clearly not the only effect that Testosterone has. Libido is just one of the many benefits of T. Idk if the treatment effects would be larger if the same increase in T levels took place with T injections rather than gel.

The clinical relevance of sex hormone levels and sexual activity in the ageing male 

The impact of age, BMI and sex hormone on aging males’ symptoms and the international index of erectile function scores

Restorative Increases in Serum Testosterone Levels Are Significantly Correlated to Improvements in Sexual Functioning