They are the same thing. I find it easier to think of graphs internal to Cat, because then the monad is just the special case of the free internal-category monad in the topos of graphs.

]]>@Mike Shulman, on this page you said virtual double categories can be defined as

```
a generalized multicategory, in the sense of Hermida, Cruttwell-Shulman, and others, relative to the monad T on graphs-internal-to-Cat whose algebras are double categories
```

did you mean “categories internal to Grph” rather than “graphs internal to Cat”? That seems to be how y’all defined it in Cruttwell-Shulman. I have a suspicion they turn out to be the same thing, but when you define the monad it is easier to think of it as categories internal to Grph because the monad is a lifting of the free category monad on Grph.

]]>I added some examples of virtual double categories that do not have composites described in Crutwell-Shulman.

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