Definition 4.2 (Cone, limit). Let $D:J\to C$ be a diagram. A cone over $D$ consists of an object $A$ (its apex) together with morphisms ${\lambda }_j:A\to D(j)$ for each $j\in \mathrm{ob}J$ (the legs of the cone) such that

commutes for each $\alpha :j\to j^{\prime }$ in $J$.

A morphism of cones $(A,({\lambda }_j|j\in \mathrm{ob}J))\to (B,({\mu }_j|j\in \mathrm{ob}J))$ is a morphism $f:A\to B$ such that ${\mu }_{j}f={\lambda }_j$ for all $j$. We have a category $Cone(D)$ of cones over $D$; a limit for $D$ is a terminal object of $Cone(D)$.

Dually, a colimit for $D$ is an initial cone under $D$.