In my post on monads and applicative functors, I used the term of “lifting to the internal hom” as the extra structure needed to make a monad an applicative functor.
As it turns out, this is better known as an enrichment on the functor or, even more generally, tensorial strength. An enrichment on a functor may only be defined for monoidal closed categories, whereas tensorial strength may also be defined for any odd monoidal category.
This goes to show how much structure we take for granted in the category of sets.