What is the word that means someone who is your arch rival like Sherlock Holmes and professor Moriarty many of the letters in the word 'enemies' are used in this word?

Probably the Word "Nemesis" is what you are looking for.
Nemesis.= (an archenemy)
It sounds like the Word you're looking for is "nemesis".
NEMESIS
I believe you can call it a "Nemesis"