A functor $T:\mathcal{C}\to\mathcal{D}$ is full if the arrow function of $T$ is surjective for every pair of objects in $\mathcal{C}$ More precisely, for every pair $C_1, C_2\in \operatorname{Ob}(\mathcal{C})$ the arrow function $T_{(C_1,C_2)}$ of $T:$ $$T_{(C_1,C_2)}:\operatorname{hom_{\mathcal{C}}}(C_1,C_2)\to\operatorname{hom_{\mathcal{D}}}(T(C_1),T(C_2))$$ given by $T_{(C_1,C_2)}(f)=T(f)$ is a surjection.