Let $\mathcal{C}$ and $\mathcal{D}$ be categories and $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors.
A natural transformation $\alpha \colon F\Longrightarrow G$ 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.
We write $\text{Nat}\webleft (F,G\webright )$ for the set of natural transformations from $F$ to $G$.
The identity natural transformation $\text{id}_{F}\colon F\Longrightarrow F$ of $F$ is the natural transformation consisting of the collection
\[ \webleft\{ \text{id}_{F\webleft (A\webright )}\colon F\webleft (A\webright )\to F\webleft (A\webright )\webright\} _{A\in \text{Obj}\webleft (\mathcal{C}\webright )}. \]
The naturality condition for $\text{id}_{F}$ is the requirement that, for each morphism $f\colon A\to B$ of $\mathcal{C}$, the diagram
commutes, which follows from unitality of the composition of $\mathcal{C}$.
Two natural transformations $\alpha ,\beta \colon F\Longrightarrow G$ are equal if we have
\[ \alpha _{A}=\beta _{A} \]
for each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$.