Hi Eduardo Ochs, Just in case you missed, internal diagrams of maps are dual to graphs (of maps) and hence called cographs (Conceptual Mathematics, pp. 293-4). As noted in the announcement of Sets for Mathematics: "The standard tools for analyzing an arbitrary map are the induced equivalence relation, co-equivalence relation, graph and cograph (cographs have been very frequently pictured in practice but only rarely recognized explicitly); all four of these are shown to arise inevitably as Kan quantifications, along the two possible interpretations of the generic map as half of the splitting of a generic idempotent" (http://www.mta.ca/~rrosebru/setsformath/; Sets for Mathematics, pp. 29-30, 80, 190). Thank you, posina On Feb 12, 2018 06:05, "Eduardo Ochs" <eduardoochs@gmail.com> wrote: Hi list, Lawvere and Schanuel use, in "Conceptual Mathematics" - from p.13 onwards - an idea that they call the "internal diagram" of a function: the internal diagram for f:A->B "looks inside" A and B and shows how each element of a A is taken to an element of B. The extension of this idea to functors is trivial: the internal diagram of a functor F:C->D shows how objects and morphisms of C are taken to objects and morphisms of D. The idea of internal diagrams for functors, natural transformations, etc, feels (to me!) as something that is not only folklore, but also something that I guess that is kept to the oral culture of courses on Category Theory - a tool that is taught informally to students as a way to help them visualize things and do the calculations, but that is practically never published in details, even in course notes... Since I started looking for these internal diagrams in CT a few months ago I stumbled on only one other place where they are mentioned - in Emily Riehl's "Category Theory in Context", in pages 17, 19, 60, and a handful of other places there. Anyone knows where else I can look for these things? All pointers will be greatly appreciated - I just wrote a section about internal diagrams for CT in these notes, http://angg.twu.net/LATEX/2017yoneda.pdf but it would be great if I could connect that to other work. Cheers, and thanks in advance, Eduardo Ochs http://angg.twu.net/math-b.html http://angg.twu.net/logic-for-children-2018.html [For admin and other information see: http://www.mta.ca/~cat-dist/ ]