Definition 4.2.15. Let $F, G : \mathcal{A} \to \mathcal{B}$ be functors. A *natural transformation*, or a *morphism of functors* $t : F \to G$, is a collection $\{ t_ x\} _{x\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})}$ such that

$t_ x : F(x) \to G(x)$ is a morphism in the category $\mathcal{B}$, and

for every morphism $\phi : x \to y$ of $\mathcal{A}$ the following diagram is commutative

\[ \xymatrix{ F(x) \ar[r]^{t_ x} \ar[d]_{F(\phi )} & G(x) \ar[d]^{G(\phi )} \\ F(y) \ar[r]^{t_ y} & G(y) } \]

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