Subsection 9.9.2 (017P): Natural Transformations

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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

\[ \webleft\{ \alpha _{A}\colon F\webleft (A\webright )\to G\webleft (A\webright )\webright\} _{A\in \text{Obj}\webleft (\mathcal{C}\webright )} \]

from $F$ to $G$ such that, for each morphism $f\colon A\to B$ of $\mathcal{C}$, the diagram

commutes.

Let $\alpha \colon F\Rightarrow G$ be a natural transformation.

For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, 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 $\text{Nat}\webleft (F,G\webright )$ for the set of natural transformations from $F$ to $G$.

Two natural transformations $\alpha ,\beta \colon F\Rightarrow G$ are equal if we have

\[ \alpha _{A}=\beta _{A} \]

for each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$.

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