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The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between finite/countable induced subgraphs extends to an automorphism of all of the Rado graph).

Q. Is there a "Rado category"?

More precisely, I'm asking if there is there a category (equivalent to a category with a countable class of morphisms) such that 1) Universality: it contains every (essentially) finite category and every category (equivalento to a category) with countable class of morphisms, up to equivalence, as full subcategories, 2) Homogeinety: every equivalence between finite/countable full subcategories extends to an autoequivalence of the category.

I'm not conversant in either category or graph theory, but I suspect the question might have to do with a 2-categorical version of the Fraissé limit. (Does it make sense?)

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    $\begingroup$ You could ask the same question of a simplicial set, semi-simplicial set, or simplicial complex. :-) $\endgroup$
    – David Roberts ♦
    Commented Oct 12, 2013 at 21:18
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    $\begingroup$ The plain Fraisse limit does the trick if you replace "equivalence" by "isomorphism". The result with equivalence sounds plausible but "partial equivalences" are much harder to make sense of than "partial isomorphism". $\endgroup$
    – François G. Dorais
    Commented Oct 12, 2013 at 22:47

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