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I added a nuts-and-bolts definition to directed graph to avoid any charges of circularity (that a directed graph is defined as a presheaf on a certain category, but that categories are certain directed graphs with extra structure).
I will note here the long-standing practice of category theorists' using the term "directed graph" to refer to what some other mathematicians call multidigraphs.
Call the things currently described at Directed graph X's.
Then we want to call X a quiver, multidigraph, diagram scheme, or as Todd has suggested, just a directed graph (with an appropriate note about the name). Then we should merge the current contents of the article at quiver into the article at free category. I do not care which of the four names we choose as primary, but we should have redirects from those names to the article, which should take the primary name. This repairs the terminology.
I added a note on terminology to directed graph, and a query under Remarks about the term "complete".
Redirects might be a good idea in the case of terms like multidigraph which are not much used in the categorical community. On the other hand, I think quiver should remain a stand-alone article, since the ways in which categorically-minded people talk about quivers and their representations is a big theory which we might want to have reflected on that page eventually (connections with ADE and so on). Descriptions of that theory would not fit very comfortably on the page path category. "Diagram scheme" probably doesn't need a separate page, but a link to concept X from diagram is a good idea if it's not already there.
I personally would vote for directed graph as the categorists' most common term for concept X.
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