\begin{definition}[Definition:Full Functor]
Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $F: \mathbf C \to \mathbf D$ be a covariant functor.
Then $F$ is '''full''' {{iff}} for all objects $C_1, C_2$ of $\mathbf C$:
:$F: \map {\operatorname{Hom}_{\mathbf C} } {C_1, C_2} \to \map {\operatorname{Hom}_{\mathbf D} } {F C_1, F C_2}, \ f \mapsto F f$
is a surjection.
Here $\operatorname{Hom}$ signifies a hom class.
\end{definition}