For any category C and any object c∈C, the constant functor Δ(c) is given by
Δ(c):J→CΔ(c)(j)=cΔ(c)(f:j→j′)=idc
Strictly speaking we have one constant functor Δ(c):J→C for each J, but usually we omit explicit reference to the source category.
By treating the Δ as a value rather than notation we obtain for each J,C a functor
Δ:C→Fun(J,C)Δ(c)=the constant functorΔ(f)j=f
The action on morphisms, which perhaps is not so clear from the definition above, is given as follows. For any f:c→c′ in C we need a morphism Δ(f):Δ(c)→Δ(c′) in the functor category; that is, we need a natural transformation Δ(f):Δ(c)⇒Δ(c′). For any j∈J, then, we need the jth component Δ(f)j to be a morphism Δ(c)(j)→Δ(c′)(j); ie, c→c′. We already have an f:c→c′ so we just use that.