Constant diagram functor (${\Delta }_J:\mathcal{C}\to [J,\mathcal{C}]$)

The constant diagram functor sends $C$ to the functor on $J$ with constant object value $C$ and identity arrow values; a morphism $f$ becomes the natural transformation with every component $f$. When all $J$-shaped categorical limits exist, it is a left adjoint to the limit functor.