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Limit (Category Theory)**

Given a (typically small) category \(\mathcal{I}\), a category \(\mathcal{C}\), and a functor \(F:\mathcal{I}\to\mathcal{C}\), a limit of \(F\) is an initial object in the category of cones over \(F\), i.e. an object \(\lim_{\mathcal{I}}F\in\mathcal{C}_0\) and a collection of arrows \(\{(\phi_i:\lim_{\mathcal{I}}F\to F(i))\in\mathcal{C}_1|i\in\mathcal{I}_0\}\), such that \(\forall (f:i\to j)\in\mathcal{I}_1:\phi_j=F(f)\circ \phi_i\), with the universal property that, whenever there is an object \(X\in\mathcal{C}_0\) with arrows \(\{(g_i:\lim_{\mathcal{I}}F\to F(i))\in\mathcal{C}_1|i\in\mathcal{I}_0\}\) such that \(\forall (f:i\to j)\in\mathcal{I}_1:g_j=F(f)\circ g_i\), there is a unique arrow \(u:\lim_\mathcal{I}F\to X\) such that \(\forall i\in\mathcal{I}_0:g_i=\phi_i\circ u\). Limits are a generalisation of products and equalisers and are dual to the concept of a colimit.