A cocone in a category \(\mathcal{C}\)
A cone in \(\mathcal{C}^{op}\)
Given a diagram \(\mathcal{J}\xrightarrow{D}\mathcal{C}\), we may take the limit of the functor \(\mathcal{J}^{op}\xrightarrow{D^{op}}\mathcal{C}^{op}\) is a cocone in \(\mathcal{C}\) - the colimit of \(D\) is this cocone.