Natural transformation

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Definition

Suppose $\mathcal{C},\mathcal{D}$ are categories, and $\mathcal{F}:\mathcal{C}\to \mathcal{D}$ and $\mathcal{G}:\mathcal{C}\to \mathcal{D}$ are functors. A natural transformation $\eta :\mathcal{F}\to \mathcal{G}$ is defined by the following data:

For every object $A\in \mathrm{O}\mathrm{b}\mathcal{C}$, a morphism ${\eta }_A\in \mathcal{D}(\mathcal{F}A,\mathcal{G}A)$,

satisfying the following compatibility condition: For any objects $A,B\in \mathrm{O}\mathrm{b}\mathcal{C}$, and any $f\in \mathcal{C}(A,B)$, we have:

$\mathcal{G}f\circ {\eta }_A={\eta }_B\circ \mathcal{F}f$.