Implication details

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

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