Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 2 / a / Solution

A categorical limit of $D:\mathcal{J}\to \mathcal{C}$ is a terminal cone $(L,(p_j))$: every cone $(X,(x_j))$ has a unique map $X\to L$ commuting with all legs.

Assume all small products and equalizers exist. PutDefine $s,t:P\rightrightarrows Q$ so that the $u:i\to j$ components areA map $X\to P$ equalizes $s,t$ exactly when its components form a cone over $D$. Therefore $\mathrm{eq}(s,t)$ represents cones and is the limit. This is the construction of small limits from products and equalizers.