colorings – What is a guarenteed amount of colors, depending on the graph’s arboricity

Your explanation is correct.

And, you can do better than $f(a) = 2a$. For example, take a complete graph on $4$ vertices: $a,b,c,d$. The arboricity is $2$ since $(a,b), (b,c),$ and $(c,d)$ forms the first tree, and the remaining edges forms a second tree. The graph is colorable with exactly four colors.