Yes, that is correct. OP is overly simplifying it. Although depression and anxiety is associated with an undersupply of serotonin, it is ultimately not the only neurotransmitter that is involved in depression. Majority of mental illnesses is not caused by an undersupply or oversupply of neurotransmitter(s) but rather an unbalanced ratio of neurotransmitters in the overall system. Like the general human body, there needs to be an equilibrium of different hormones, nutrients, mineral, etc for normal physiological processes to occur. If there isn't, that's when the body starts to go all wonky and you have certain conditions. And as /u/nicetomeetyoufriend mentioned, there are different subtypes of receptors for specific neurotransmitters and each subtype has a different and specific function. In mental illnesses, there could also be an unequal distribution of these receptions throughout the system which limits the effects of neurotransmitters. Even to this day, the physiological mechanisms are still unknown when it comes to mental illnesses and we tend to simplify them as an undersupply/oversupply of an neurotransmitter to the general public to understand, but in reality, there is more to mental illnesses than just neurotransmitters.