If we now pass more generally to any -enriched category then we still have the enriched hom object functor . One says that is powered over if it is in addition equipped also with a mixed operation such that behaves as if it were a hom of the object into the object in that it satisfies the natural isomorphism
We say that is powered or cotensored over if all such power objects exist.
Powers are frequently called cotensors and a -category having all powers is called cotensored, while the word “power” is reserved for the case Set. However, there seems to be no good reason for making this distinction. Moreover, the word “tensor” is fairly overused, and unfortunate since a tensor (= a copower) is a colimit, while a cotensor (= power) is a limit.
Powers are a special sort of weighted limits. Conversely, all weighted limits can be constructed from powers together with conical limits. The dual colimit notion of a power is a copower.