CatDat

• Notation: $F: \Grp \to \Set$
• Domain: category of groups
• Codomain: category of sets

This functor maps a group $G$ to the collection $$F(G) := \textstyle\prod_{\kappa} \Hom(L_\kappa,G),$$ where $\kappa$ ranges over all infinite cardinals and $L_\kappa$ is an infinite simple group of cardinality $\kappa$; we choose $L_\kappa := \Alt_{\fin}(\kappa)$ to make things concrete. For each $G$, the collection $F(G)$ is isomorphic to a set, since for $\kappa > \card(G)$ every homomorphism $L_\kappa \to G$ is trivial. Thus, a more precise definition would be $$F(G) := \textstyle\prod_{\kappa \leq \card(G)} \Hom(L_\kappa,G),$$ but the first definition makes it more apparent that $F$ is a functor. This is the canonical example of a continuous functor $\Grp \to \Set$ that is not representable, and not a right adjoint.