Consider the diagram of graphs...
Solution.
We’re given a “diagram of graphs” with the shape of the bipointed object below:
I’m thinking that the idea here is replace the objects representing our domain and codomain with “dots”, so we can treat the maps like “arrows”. There should be unique maps to the terminal object \(P \rightarrow T, B_1 \rightarrow T,
B_2 \rightarrow T\) that have the following structure:
All together, there are 5 objects in the diagram: 3 dots referring to \(P,B_1,B_2\) respectively, and two arrows referring to the respective projection maps. I’m going to use this to construct a table of elements and their respective source and target maps by using self referential arrows for the points:
| Object | Description | Source | Target |
| 0 | \(P \rightarrow T\) | 0 | 0 |
| 1 | \(B_1 \rightarrow T\) | 1 | 1 |
| 2 | \(B_2 \rightarrow T\) | 2 | 2 |
| 3 | \(P \rightarrow B_1\) | 0 | 1 |
| 4 | \(P \rightarrow B_2\) | 0 | 2 |
The two arrows in the diagram represent the two different maps from \(P\text{,}\) so we can map from \(A\) to the above diagram in two distinct ways:
These two maps \(m_1,m_2\) are “incident” at the one dot that represented \(P\) in our original diagram. We can name these by referring to the “source” and “target” maps which are the only two maps \(D \rightarrow A\text{:}\)
This means we can express the incidence relation by the equation \(m_1 s = m_2 s\text{,}\) which in the categorgy \(\mathcal{S}^{\downarrow_\bullet^\bullet \downarrow}\) should imply \(s' m_1 = s' m_2\) or basically asserting the commutativity of the diagram below:
I think I’m on the right track here, but I’m not quite sure how it fits into the big picture yet.
