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Equaliser**

Given a pair of parallel arrows \(f,g:X\to Y\) in a category \(\mathcal{C}\), their equaliser is an object \(\mathrm{Eq}(f,g)\in\mathcal{C}_0\) with an arrow \(u:\mathrm{Eq}(f,g)\to X\) such that \(f\circ u=g\circ u\) and satisfying the universal property that, whenever there is an arrow \(h:Z\to X\) such that \(f\circ h=g\circ h\), there is a unique arrow \(\bar{h}:Z\to\mathrm{Eq}(f,g)\) such that \(h=u\circ \bar{h}\) - any map which makes \(f,g\) equal upon precomposition factors through the equaliser. This is an example of a limit in category theory. In \(\mathbf{set}\), the equaliser of two maps \(f,g:X\to Y\) is the subset \(\mathrm{Eq}(f,g)=\{x\in X|f(x)=g(x)\}\) and the arrow \(u\) is just its inclusion into \(X\).