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Given parallel functors \(F,G:\mathcal{C}\to\mathcal{D}\) between categories \(\mathcal{C},\:\mathcal{D}\), a 'natural transformation' \(\eta:F\Rightarrow G\) is an assignment of an arrow \((\eta_x:F(x)\to G(x))\in\mathcal{D}_1\) to each object \(x\in\mathcal{C}_0\), in a 'natural way', i.e. for every arrow \((f:x\to y)\in\mathcal{C}_1\) there is a commutative square in \(\mathcal{D}\): \(\eta_y\circ F(f)=G(f)\circ \eta_x\). Natural transformations (combined with vertical composition turn the hom-set \(\mathbf{Cat}(\mathcal{C},\mathcal{D})\) of functors from \(\mathcal{C}\) to \(\mathcal{D}\) into a category of its own, while vertical and horizontal composition together turn \(\mathbf{Cat}\) into a 2-category.