Implication Details

Claim: A functor is an equivalence if and only if it is left-invertible and is right-invertible.

Proof: If a functor $F$ has a right inverse $R$ and a left inverse $L$ , then $L \cong L \circ F \circ R \cong R.$ Hence, $R$ (and $L$ ) are (pseudo-)inverse to $F$ .

Show 17 functors using this implication

binary coproduct functor on sets
empty functor to the category of sets
forgetful functor from finite sets to sets
forgetful functor from finite abelian groups to abelian groups
forgetful functor from finite groups to groups
forgetful functor from groups to pointed sets
free group functor
identity functor on the category of sets
inclusion functor from extended natural numbers to ordinal numbers
morphism endpoints inclusion
opposite category functor
opposite monoid functor
contravariant power set functor
covariant power set functor
span endpoints inclusion
trivial functor from the category of groups
walking isomorphism object inclusion