In p.223 of Conceptual Mathematics, 2nd ed., by Lawvere and Schanuel, Exercise 21 asks:
If $A, D$ denote the generic arrow and the naked dot in $S^{\Downarrow}$, show that
$$ A \times A = A + D +D $$
Hint: Besides counting the arrows and dots of an arbitrary graph $X$ (such as $X = A \times A$) via maps $A \rightarrow X$, $D \rightarrow X$, the actual internal structure of $X$ can be calculated by composing these maps with the two maps $s,t: D \rightarrow A$.
For context:
$S^{\Downarrow}$ is the category of irreflexive graphs, whose objects are two sets $X$ (the set of arrows) and $P$ (the set of points) with maps $source: X \rightarrow P$ and $target: X \rightarrow P$. A map between graphs is a structure-preserving morphism with respect to these two graphs.
$A$ the generic arrow is the graph that looks like * -> *.
$D$ the naked dot is the graph that looks like *.
This problem appears in the section for the distributive law which states the following maps are isomorphisms:
$$ (A \times B) + (A \times C) \rightarrow A \times (B + C) $$ $$ 0 \rightarrow A \times 0 $$
My interpretation of the hint is that any graph map $A \rightarrow X$ is equivalent to a graph map $D \rightarrow X$ defined as the arrow map pre-composed with either the $s$ or $t$ map, since every arrow must have either a source or target. Beyond that I have been having trouble figuring this out.. I assume the use of the distributive law is needed at some point.. maybe using something like
$$ A \times A = A \times (A + 0) $$
or
$$ A + D + D = A + (D \times (1 + 1)) $$
Any guidance would be appreciated.
Call $a,b$ the vertices of $A$, i.e. $A$ is the graph $a\to b$.
Then the vertices of $A\times A$ are $(a,a),\ (a,b),\ (b,a),\ (b,b)$, and we have only one arrow $(a,a)\to (b,b)$ and thus $(a,b)$ and $(b,a)$ are isolated points.
We can formulate then a proof according to the hint, purely categorically, as follows:
We have to prove that $B:= A+D+D$ (depicted as $\left[\matrix{a&d_1\\d_2&b}\right]$ with an arrow $a\to b$) is a product of $A$ and $A$.
Consider the graph homomorphisms $\pi_1:a,d_1\mapsto a;\ d_2,b\mapsto b$ and $\pi_2:a,d_2\mapsto a;\ d_1,b\mapsto b$, and prove the universal property.