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A 2-Category \(\mathcal{C}\) is a category which is enriched over \(\mathbf{Cat}\), i.e for each pair of objects \(x,y\in\mathcal{C}_0\), the hom-set \(\mathcal{C}(x,y)\) forms a category, i.e. there is a collection \(\mathcal{C}_2\) of 2-arrows, each of which has a domain and codomain a pair of parallel 1-arrows in \(\mathcal{C}_1\), and these behve nicely under composition of 1-arrows. In particular, for 1-arrows \(f,g:X\to Y\) and \(h,k:Y\to Z\), with 2-arrows \(\eta:f\Rightarrow g\) and \(\xi:h\Rightarrow k\), there is a composite 2-arrow \((\xi*\eta):(h\circ f)\Rightarrow (k\circ g)\).