I have the following pair of graphs.
[Judith L. Gersting Book]
Let $G$ and $H$ be the graphs in the image. For $G$, $1$ and $4$ have degree 3, while for $H$, $c$ and $d$ have degree 3. The rest have degree $2$. I need to map $\{1, 4\}$ to $\{c, d \}$ and $\{2, 3, 5, 6 \}$ to $\{a, b, e, f \}$. The function $f(1) = c, f(2) = a, f(3) = b, f(4) = d, f(5) = e, f(6) = f$ seems to work. How can I prove it preservers the arc relation?
You've got several options. One would be to just exhibit an isomorphism that works, as you did. Another, as saulspatz suggests in the comments, is to illustrate that graphically by drawing the graph on the right labeled with the vertices 1-6 as per your isomorphism. The strategy I'll add is to describe both graphs in plain English, like "two 3-cycles joined by a single edge", with the underlying assumption being that that description is an invariant.
Honestly, they all have the same drawback, which is that anyone checking your work for mistakes (or typos) has to hunch over both graphs and double-check the work edge by edge. I'll humbly suggest that my strategy is the easiest to verify and lets the reader construct the isomorphism in their mind.