Let 
be a category.
Then 
is said to be a subcategory of 
, if the objects of 
are also objects of 
, if the morphisms of 
are also morphisms of 
, and if 
is a category in its own right. In particular, for each object
of 
, the identity morphism 
from 
should also be in 
, and compositions of morphisms in 
should also be in 
.
The functor denoted by 
,
the so-called inclusion functor which sends objects identically
to objects, and morphisms identically to morphisms, is always a faithful
functor.
