Subfunctor

Let ${\displaystyle {\mathcal {C}}}$ be a category, and let ${\displaystyle F}$ be a contravariant functor from ${\displaystyle {\mathcal {C}}}$ to the category of sets Set. A contravariant functor ${\displaystyle G}$ from ${\displaystyle {\mathcal {C}}}$ to Set is a subfunctor of F if

1. For all objects c of ${\displaystyle {\mathcal {C}}}$, ${\displaystyle G(c)\subseteq F(c)}$, and
2. For all arrows ${\displaystyle f:c'\rightarrow c}$ of ${\displaystyle {\mathcal {C}}}$, ${\displaystyle G(f)}$ is the restriction of ${\displaystyle F(f)}$ to ${\displaystyle G(c')}$.

This relation is often written as ${\displaystyle G\subseteq F}$.

For example, let 1 be the category with a single object and a single arrow. A functor F: 1 → Set maps the unique object of 1 to some set S and the unique identity arrow of 1 to the identity function 1_S on S. A subfunctor G of F maps the unique object of 1 to a subset T of S and maps the unique identity arrow to the identity function 1_T on T. Notice that 1_T is the restriction of 1_S to T. Consequently, subfunctors of F correspond to subsets of S.