Diagonal functor

Functor

 context ${\bf C},{\bf D}$ … category definiendum $\Delta:{\bf C}\longrightarrow{\bf C}^{\bf D}$ definition $\Delta C:=\Delta_C$ definition $\Delta(f):=\mathrm{const}_f$

Discussion

Coherence

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.