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Difference between Come up against someone and Run up against someone

come up against someone—(also: go up against someone) challenge a competitor in a contest, election, etc:

  • Although he rode very well, he came up against an opponent who tactically rode even better.

run up against someone

1. meet a person, especially by accident:

  • You can’t go anywhere in London without running up against him.

2. = come up against someone:

  • I also had the sense that Tony would be a formidable adversary if you ran up against him in court.