Definition 1.11 (Faithful / full / essentially surjective). Let $F:C\to D$ be a functor.

• (a)
  We say $F$ is faithful if, given $f$ and $g$ in $\mathrm{mor}C$, $(Ff=Fg$, $\mathrm{dom}f=\mathrm{dom}g$, $\mathrm{cod}f=\mathrm{cod}g)⟹f=g$.
• (b)
  We say $F$ is full if, for every $g:FA\to FB$ in $D$, there exists $f:A\to B$ in $C$ with $Ff=g$.
• (c)
  We say $F$ is essentially surjective if, for any $B\in \mathrm{ob}D$, there exists $A\in \mathrm{ob}C$ with $FA\cong B$.

Note that if $F$ is full and faithful, it’s essentially injective: given in $D$, the unique $A\stackrel{f}{\to }B$ with $Ff=g$ is an isomorphism.

We say $D\subseteq C$ is a full subcategory if the inclusion $D\to C$ is a full functor.