underlying functor

underlying functor (plural underlying functors)

1. (category theory) a forgetful functor
   • 1995, Michael Barr with Charles Wells, Category Theory for Computing Science‎^[1], 2nd edition, University Press, Cambridge, Great Britain: Prentice Hall, §3.1.10, page 57:

     3.1.10 Example  If you forget [that] you can compose arrows in a category and you forget which arrows are the identities, then you have remembered only that the category is a graph. This gives an underlying functor ${\displaystyle U:\mathbf {Cat} \rightarrow \mathbf {Grf} }$ , since every functor is a graph homomorphism although not vice versa.