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Functor**

Given categories \(\mathcal{C},\:\mathcal{D}\), a (covariant) functor \(F:\mathcal{C}\to\mathcal{D}\) is a pair of maps (typically both simply labelled \(F\)) \(F_0:\mathcal{C}_0\to\mathcal{D}_0\) and \(F_1:\mathcal{C}_1\to\mathcal{D}_1\) which are compatible with identities and compositions, i.e. \(\forall X\in \mathcal{C}_0:F_1(1_X)=1_{F_0(X)}\) and \(\forall X\overset{f}{\to}Y\overset{g}{\to}Z\in\mathcal{C}:F_1(g\circ f)=F_1(g)\circ F_1(f)\).