Editing Full and faithful functors (section)

==Formal definitions==
Explicitly, let ''C'' and ''D'' be ([[locally small category|locally small]]) [[category (mathematics)|categories]] and let ''F'' : ''C'' → ''D'' be a functor from ''C'' to ''D''. The functor ''F'' induces a function
:<math>F_{X,Y}\colon\mathrm{Hom}_{\mathcal C}(X,Y)\rightarrow\mathrm{Hom}_{\mathcal D}(F(X),F(Y))</math>
for every pair of objects ''X'' and ''Y'' in ''C''. The functor ''F'' is said to be
*'''faithful''' if ''F''<sub>''X'',''Y''</sub> is [[injective]]<ref>Mac Lane (1971), p. 15</ref><ref name="Jacobson-09">Jacobson (2009), p. 22</ref>
*'''full''' if ''F''<sub>''X'',''Y''</sub> is [[surjective]]<ref name="Jacobson-09"/><ref>Mac Lane (1971), p. 14</ref>
*'''fully faithful''' (= '''full and faithful''') if ''F''<sub>''X'',''Y''</sub> is [[bijective]]
for each ''X'' and ''Y'' in ''C''.