CatDat

Implication Details

Claim: Given a functor whose codomain is trivial, and whose domain is strongly connected, then it is full.

Proof: Let $\C$ be a strongly connected category. Then the unique functor $F : \C \to 1$ to the trivial category is full: for all $X,Y \in \C$ the map $\Hom(X,Y) \to \Hom(F(X),F(Y))$ is surjective since its domain is non-empty and its codomain is a singleton set.