Presheaf

Remarks

Example: let $T$ be a topological space and let $\mathcal{O}(T)$ be the category whose objects are the open sets of $T$, with one morphism $V \to U$ when $V \subseteq U$. Let $F(U)$ be the set of continuous functions $U \to \R$ and let $F(V \to U)$ be restriction $s \mapsto s|_V$. Restriction to $U$ is the identity and restricting to $V$ then to $W$ equals restricting to $W$, so $F$ is a presheaf.

Non-example: on $\mathcal{O}(T)$, let $G(U)$ be the set of all functions $U \to \R$ and send $V \subseteq U$ to extension by zero $G(V) \to G(U)$. This is a functor $\mathcal{O}(T) \to \mathbf{Set}$, but its maps go from $V$ to $U$, so it is covariant and not a presheaf on $\mathcal{O}(T)$.

Every hom functor $\Hom(-, A)$ is a presheaf, and a presheaf on $\mathcal{O}(T)$ is a presheaf on $T$ in the sense of sheaf theory.