For a small category $\mathsf{J}$, define a functor $i_0 \colon \mathsf{J} \to \mathsf{J} \times \mathbbe{2}$ so that the pushout \(\xymatrix{ \cJ \ar[d]_{i_0} \ar[r]^-{!} \ar@{}[dr]\mid(.8){\displaystyle\ulcorner} & \mathbbe{1} \ar[d]^s \\ \mathsf{J} \times \mathbbe{2} \ar[r] & \mathsf{J}^{\triangleleft}}\) in $\textup{\textsf{cat}}$ defines the cone over $\mathsf{J}$, with the functor $s \colon \mathbbe{1} \to \mathsf{J}^\triangleleft$ picking out the summit object. Remark \ref{rmk:limit-diagram-shape} gives an informal description of this category, which is used to index the diagram formed by a cone over a diagram of shape $\mathsf{J}$.