Let $D\colon \mathcal{J} \to \mathcal{C}$ be a [[Diagrams |diagram]] in $\mathcal{C}$. A **cone** to the diagram $D$ with **apex** (or “tip”) $N \in \mathrm{Ob}(\mathcal{C})$ is given by: 1. An object $N$ in $\mathcal{C}$ (the apex). 2. A family of morphisms $\{\pi_j : N \to D(j)\}_{j \in \mathcal{J}}$ (one arrow for each object $j$ of $\mathcal{J}$) such that for **every** morphism $\alpha : j \to j'$ in $\mathcal{J}$, the following triangle in $\mathcal{C}$ commutes: $D(\alpha) \circ \pi_j \;=\; \pi_{j'}$. This commuting condition means intuitively that the family $\{\pi_j\}$ is “compatible” with all the structure (arrows) of the diagram $D$. We say “$N$ is a **cone** over $D$ with legs $\{\pi_j\}$.” To summarize the relationship between diagrams, cones, and limits: - A **diagram** is a functor $D : \mathcal{J} \to \mathcal{C}$. - A **cone** to $D$ with apex $N$ is a natural way of “landing on” $D$ from a single object $N$ in $\mathcal{C}$. - A **limit** is a cone that is “optimal” or **universal**, meaning every other cone to the same diagram factors uniquely through it. These concepts unify many familiar constructions in mathematics (products, pullbacks, inverse limits, etc.) under one categorical perspective.