The category of (small) categories and functors is the category
- Objects. The objects of
are small categories. - Morphisms. For each
, we have - Identities. For each
, the unit mapof
at is defined bywhere
is the identity functor of of Example 9.5.1.1.4. - Composition. For each
, the composition mapof
at is given bywhere
is the composition of and of Definition 9.5.1.1.5.