Implication Details

Assumptions: one-way

Proof: Every reflexive pair is equal: If $f s = g s = \id$ , then since $s f = \id$ (one-way), we must have $f = s^{-1}$ , and likewise $g = s^{-1}$ .

Show 10 categories using this implication

empty category
trivial category
discrete category on two objects
category of free abelian groups
category of smooth manifolds
category of metric spaces with continuous maps
poset of extended natural numbers
category of sets and relations
category of schemes
category of sets with finite-to-one maps