Constant Diagram Functor

Let \(F: J \to C\) be a diagram on \(C\) of shape \(J\). For any \(c \in C\), one may consider the constant functor at \(c\) to be the functor \(\Delta c: J \to C\) such that every element of \(J\) is mapped to \(c\). The constant diagram functor \(\Delta: C \to C^J\) is the functor such that \(\Delta: c \mapsto \Delta c\), where morphisms \(f: c \to c'\) in \(C\) are mapped to the constant natural transformation \(\Delta f: \Delta c \Rightarrow \Delta c'\).

Cones

Constant diagram functors have particular relevance in the study of cones and limits. A cone over a diagram \(F: J \to C\) with summit \(c \in C\) is a natural transformation \(\lambda: \Delta c \Rightarrow F\), which is to say a natural transformation whose domain is a constant functor at the summit.