Using the same factor mapping for multiple columns in R

I'm trying to figure out how to use the same factor "mapping" for more than one column.

For example, here's a data set of imaginary tennis matches, in which one player was the favorite and the other player was the underdog:

df <- tribble(
  ~game, ~favorite, ~underdog,
  #-----|----------|----------
  1,     "Mary",     "Susan",
  2,     "Susan",    "John",
  3,     "Mary",     "Joe",
  4,     "Joe",      "Dave",
  5,     "Susan",    "Dave"
)

Note that there are favorites who are never underdogs (Mary) and underdogs who are never favorites (John, Dave).

I'd like to create a factor that works for both the favorite and underdog columns. For instance, if I now enter

df %<>% mutate(favorite=factor(favorite))

then Joe is 1, Mary is 2, and Susan is 3, since the default is alphabetical order.

How can I now change the underdog column to a factor so that every time Joe shows up there he is assigned 1, Mary gets 2, and so on?

My requirements/hopes/desires:

1. In practice there will be a lot of rows and different entries, so hand coding is not an option;
2. I'd like to be able to do this in the middle of a chain, with pipes;
3. At the end, I'd like to be able to run code like filter(favorite=="Susan" | underdog=="Susan").

I looked but couldn't find where this has been done before. Any help would be appreciated.