Consider a character has proficiency using the net. They also have the Net Adept feat from UC. Now this feat allows the character to use the neat as a reach melee weapon. It does not, however, determine the class (light, martial) of the weapon used in melee.
Is 6lbs considered light enough to be used off-hand in a TWF build?
Net Adept (Combat)
You have trained to use the net as a melee weapon.
Prerequisites: Exotic Weapon Proficiency (net), base attack bonus +1.
Benefit: You can treat a net as a one-handed melee reach weapon with a 10-foot reach. Further, you take no penalty on melee attack rolls for using an unfolded net, and you can use one full-round action or two move actions to fold a net.
Normal: A net is a ranged weapon that imposes a –4 penalty on ranged attack rolls if it is unfolded. Folding a net takes a proficient user 2 rounds
It's classified as one-handed. As in "light, one-handed, two-handed", not just giving you how many hands you need to wield it.