In what ways is "The Red-headed League" a Manichean text?
You can make the argument that all mysteries and crime stories are examples of Manichaeism, because all detective stories can be seen as a battle between good and evil.
Manichaeism can be defined as "the struggle between a good, spiritual world of light, and an evil, material world of darkness" (Wikipedia).
In “The Red-headed League,” Holmes is battling against a man he has fought before: John Clay. This makes the man a nemesis, and a representation of evil, making Holmes the representation of good.
He is, in my judgment, the fourth smartest man in London, and for daring I am not sure that he has not a claim to be third. I have known something of him before. (p. 13)
Although Clay is young, he is smart and daring and described as a “murderer, thief, smasher, and forger” (p. 15). Apparently he is very clever and at the top of his game, making him a worthy opponent.
Another indication of Manichaeism is the end of the story, when Holmes and Clay congratulate each other. Clearly each respects the other as a symbol.