Let $\mathcal{C}$ and $\mathcal{D}$ be categories and $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors.
A natural transformation $\alpha \colon F\Rightarrow G$ from $F$ to $G$ is a transformation
from $F$ to $G$ such that, for each morphism $f\colon A\to B$ of $\mathcal{C}$, the diagram
Let $\alpha \colon F\Rightarrow G$ be a natural transformation.
For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the morphism $\alpha _{A}\colon F_{A}\to G_{A}$ is called the component of $\alpha $ at $A$.
We denote natural transformations such as $\alpha $ in diagrams as
We write $\operatorname {\mathrm{Nat}}\webleft (F,G\webright )$ for the set of natural transformations from $F$ to $G$.
Two natural transformations $\alpha ,\beta \colon F\Rightarrow G$ are equal if we have
for each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$.
