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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