great diagram Michael – but maybe the arrows \$$c\rightarrow c'\$$ and \$$L(c)\rightarrow L(c')\$$ should be pointing the other way round?

that way the diagram would also show how the maps \$$\mathcal{C}(c, R(d))\rightarrow \mathcal{C}(c', R(d'))\$$ and \$$\mathcal{D}(L(c), d)\rightarrow \mathcal{D}(L(c'), d')\$$ work.