Let $\mathcal{C}$ and $\mathcal{D}$ be categories. A constant functor from $\mathcal{C}$ to $\mathcal{D}$ is a functor $k:\mathcal{C\to D}$ such that there is an object $A\in \mathcal{D}$ such that

• for all objects $X$ in $\mathcal{C}$ , $k(X)=A$ , and
• for all morphisms $X\to Y$ in $\mathcal{C}$ , $k(X\to Y)=1_A$ , the identity morphism of $A$ .

Remarks.

• For the constant functor $k$ considered above, the object $A$ is the fixed value of $k$ . To identify $k$ with $A$ , we often write $k_A:=k$ , or simply $A$ when no confusion arises.
• Composing a functor with a constant functor gives us a constant functor. More precisely, let $k_A:\mathcal{C\to D}$ be the constant functor with fixed value $A$ . If $F:\mathcal{B\to C}$ is a functor, then $k_A\circ F:\mathcal{B\to D}$ is the constant functor $(k\circ F)_A$ with fixed value at $A$ . Moreover, if $G:\mathcal{D\to E}$ is a functor, then $G\circ k_A:\mathcal{C\to E}$ is the constant functor $(G\circ k)_{G(A)}$ valued at $G(A)$ .
• Given any functor $T:\mathcal{C\to D}$ , any natural transformation $\tau_A: k_A\dot{\to} T$ takes any object $X\in \mathcal{C}$ to a morphism $A\to T(X)$ , and, for any morphism $\alpha: X\to Y$ in $\mathcal{C}$ , a commutative triangle
  
  in $\mathcal{D}$ . Similarly, any natural transformation $\sigma_A:T\dot{\to} k_A$ takes any object $X$ to $T(X)\to A$ and any morphism $\alpha: X\to Y$ to a commutative triangle
  
• If $\alpha:A\to B$ is any morphism in $\mathcal{D}$ , then the natural transformation $\tau:k_A\dot{\to} k_B$ given by sending every object $X\in \mathcal{C}$ to the morphism $\tau_X: A\to B:=\alpha$ can be thought of as a ``constant'' natural transformation, since, for any morphism $\beta:X\to Y$ in $\mathcal{C}$ , the commutative diagram
  
  reduces to $\alpha$ .
• As above, denote $\tau$ by $\tau_{\alpha}$ . Let $\sigma_A: T\dot{\to} k_A$ be a natural transformation. Then the transformation $\tau_{\alpha}\circ \sigma_A:T\dot{\to} k_B$ sends any object $X$ in $\mathcal{C}$ to the morphism $$T(X)\to A \stackrel{\alpha}{\to} B$$ in $\mathcal{D}$ , and any morphism $\beta:X\to Y$ to the commutative diagram
  
  For any natural transformation $\gamma_B: k_B\dot{\to} T$ , the composition $\sigma_B\circ \tau_{\alpha}$ works similarly.

Bibliography

1

S. Mac Lane, Categories for the Working Mathematician (2nd edition), Springer-Verlag, 1997.