A fully faithful functor 
is a functor for which
the induced function
 |
is a bijection for every pair of objects 
of 
. A faithful functor requires
these functions only to be injections, while a full
functor requires them to be surjections. Therefore
fully faithful means both full and faithful; every fully faithful functor is faithful,
but the converse need not hold. Thus a fully faithful functor preserves all morphisms
between objects and introduces no additional morphisms
between their images.
