Let \(J\) be a small category, \(\mathcal C\) a category, and \(F\colon J\to\mathcal C\) a functor. A cone over \(F\) is an object \(Y\) of \(\mathcal C\) with morphisms \(\psi_j\colon Y\to F(j)\) for \(j\in J\) such that \(F(u)\circ\psi_j=\psi_k\) for every morphism \(u\colon j\to k\) in \(J\). A limit of \(F\) is a cone \((L,(\pi_j)_{j\in J})\) over \(F\) such that for every cone \((Y,(\psi_j)_{j\in J})\) over \(F\) there is a unique morphism \(h\colon Y\to L\) with \(\pi_j\circ h=\psi_j\) for all \(j\in J\).
Products are limits over a category \(J\) with no nonidentity morphisms, pullbacks are limits over \(\bullet\to\bullet\leftarrow\bullet\), and inverse limits are limits over a directed set, viewed as a category with a morphism \(\beta\to\alpha\) whenever \(\alpha\leq\beta\).
The dual notion, obtained by reversing every morphism, is the colimit. If every functor \(J\to\mathcal C\) has a limit, then \(\lim\colon\mathcal C^J\to\mathcal C\) is a right adjoint of the diagonal functor \(\Delta\colon\mathcal C\to\mathcal C^J\), which sends an object to the constant functor at it.