Let
be a category. The identity functor of
is the unique functor, written
, such that for every object and every morphism in
, we have
and
To verify that
is indeed a functor, we note that
, where is the identity morphism of , and
.
For any functor
, we have
.
Since every category gives rise to its unique identity functor, we can think of the identity functor as a (covariant) functor on Cat, the category of (small) categories. It is given by taking any category
to itself and any functor
to itself.
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and 
