nLab
constant functor

Definition

A constant functor $\Delta (d):C\to D$ is a functor that maps each object of the category $C$ to a fixed object $d\in D$ and each morphism of $C$ to the identity morphism of that fixed object.

Note that a constant functor can be expressed as the composite

$$C\stackrel{!}{\to }1\stackrel{[d]}{\to }D.$$C \stackrel{!}{\to} 1 \stackrel{[d]}{\to} D.

Here $1$ is a terminal category (exactly one object and exactly one morphism, namely the identity), and $[d]$ denotes the unique functor from $1$ with $F(•)=d$ and $F({\mathrm{Id}}_{•})={\mathrm{Id}}_d$.

Examples

• For $F$ any functor, a natural transformation

  ${\Delta }_d\Rightarrow F$

  from a constant functor into $F$ is precisely a cone over $F$. Similarly are natural transformation

  $F\Rightarrow {\Delta }_d$

  is a cocone.

Related concepts

• constant sheaf
• diagonal functor