| context | ${\bf C},{\bf D}$ … category |
| definiendum | $\Delta:{\bf C}\longrightarrow{\bf C}^{\bf D}$ |
| definition | $\Delta C:=\Delta_C$ |
| definition | $\Delta(f):=\mathrm{const}_f$ |
Note that for constant functors $\Delta_A,\Delta_B$, the square in the definition of a natural transformation commutes trivially. Hence any arrow $f:{\bf C}[A,B]$ gives rise to a natural transformation.
Wikipedia: Diagonal functor
nLab: Diagonal functor