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.