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categories: Re: categories or graphs?
Diagrams may sometimes be graphs. They may sometimes be categories,
but it's pretty rare to specify the identity maps in them.
So surely the right definition is that they are "elementary sketches",
ie category presentations, not that there is any need to generate
the category from them.
This is the view that I take in Section 7.3 of "Practical Foundations".
http://www.dcs.qmul.ac.uk/~pt/Practical_Foundations/html/s73.html
Paul