For example.It seems we can say
"He has no equal for running",
but "He has no equal to running."
Meanwhile,it also seems we can say "He haven't the key to the door", but "He haven't the key for the door."
He has no equal in running. ) He hasn't the key to the door. ) He hasn't the key for the door.
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He has no equal in running. (He is better than everyone else in the sport of running.)
He hasn't the key to the door. (He does not have a key that fits in the lock of the door.)
He hasn't the key for the door. (He does not have a key for the purpose of unlocking the door.)
Usually "to" is used for abstract nouns.
He has