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Mar 14, 2021 at 4:14 comment added David Kern The fact that this path functor $F$ is comonadic is explained in detail in the "Prologue" to Dawson–Paré–Pronk's Paths in double categories. In fact the "extra structure" making a category (free on) a graph is a property, which comes from the fact that the $2$-comonad induced by the extension of $F$ to a $2$-functor from (op)virtual double categories to double categories is oplax-idempotent (Theorem 1.21 op.cit.)
Feb 24, 2021 at 17:06 comment added Alec Rhea This is all very cool, thank you!
Feb 24, 2021 at 15:22 history edited Tim Campion CC BY-SA 4.0
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Feb 24, 2021 at 15:06 history edited Tim Campion CC BY-SA 4.0
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Feb 24, 2021 at 14:52 history edited Tim Campion CC BY-SA 4.0
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S Feb 24, 2021 at 14:46 history answered Tim Campion CC BY-SA 4.0
S Feb 24, 2021 at 14:46 history made wiki Post Made Community Wiki by Tim Campion