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I’ve added to reflexive graph a definition of the free category of a reflexive quiver.
That page needs some reorganization because everything now said there is about reflective quivers, and not say about reflective undirected simple graphs.
Maybe free category also also needs touching up and maybe a link to reflective graph. I don’t know how to justify that the paths in the free category don’t contain identity edges.
That is indeed one way of defining the free category on a reflexive quiver, but another way of putting it is summarized like this: take the free/path category on the underlying quiver, and then take the evident quotient category of that by dividing by the smallest categorical congruence which makes the chosen loops equivalent to identity arrows. The description you gave can be regarded as giving a normal form for each equivalence class of paths.
(The normal form description may look simpler, but the uses of ’not’ and ’non’ make it less well-adapted to internalization inside other categories.)
There’s hardly any “reorganization” that needs to be done because the article is already so short to begin with. The monadicity of reflexive quivers over $Set$ is a useful stand-alone result that doesn’t need any fixing (one thing that I think was probably said elsewhere, maybe at monadic functor, is that this underlies an example where monadic functors do not compose: $Cat$ is monadic over reflexive quivers, and reflexive quivers are monadic over $Set$, but $Cat$ is not monadic over $Set$).
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