Categories

For the definition, see Wikipedia.

Here, at this time, we just want to give a small diagram that shows some of the relations between the categories of categories, monoids, (directed) graphs, and sets.
\[\begin{array}{cccccc} \Mon& \subset & \Cat\\ \downarrow && \downarrow\\ \Set & \subset & \Graph\\ \end{array}\] Here the vertical arrows take, respectively, a monoid to its underlying set and a category to its underlying graph (i.e., they forget the composition and the specification of the identities).
The top inclusion takes a monoid into a category with just one object and that monoid as its set of arrows.
The bottom inclusion takes a set into the graph with only one vertex and that set as its set of edges.

Both vertical (downward) arrows have left adjoints going up, taking respectively a set to the free monoid on that set and a graph to the free category on that graph.

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Following is work in progress!
Here are two diagrams, one for ordinals and the other for internal categories:

$\begin{array}{} &&&& \{1\} \\ &&& \swarrow && \searrow \\ && \{0,1\} &&&& \{1,2\} \\ & \nearrow && \searrow && \swarrow && \nwarrow \\ &&&& \{0,1,2\} \\ &&&& \uparrow \\ \{0\} && \rightarrow && \{0,2\} && \leftarrow && \{2\} \\ \end{array}$

$\begin{array}{} &&&& \catC_0 \\ &&& \nearrow && \nwarrow \\ && \catC_1 &&&& \catC_1 \\ & \swarrow && \nwarrow && \nearrow && \searrow \\ &&&& \catC_2 \\ &&&& \downarrow \\ \catC_0 && \leftarrow && \catC_1 && \rightarrow && \catC_0 \\ \end{array}$