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 <equalizer>[equalizers] exist. Put
$$
P=\prod_{j\in\mathcal J}D(j),
\qquad
Q=\prod_{u:i\to j}D(j).
$$
Define $s,t:P\rightrightarrows Q$ so that the $u:i\to j$ components are
$$
s_u=D(u)\pi_i,\qquad t_u=\pi_j.
$$
A map $X\to P$ equalizes $s,t$ exactly when its components form a cone over $D$. Therefore $\operatorname{eq}(s,t)$ represents cones and is the limit. This is the <construction of small limits from products and equalizers>.

Solution

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

If all small products and equalizers exist, the limit of $D:\mathcal J\to\mathcal C$ is the equalizer of the two maps
$$
\prod_{j\in\mathcal J}D(j)\rightrightarrows
\prod_{u:i\to j}D(j),
$$
whose $u$-coordinates are respectively $D(u)$ after projection to $D(i)$ and direct projection to $D(j)$.


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 Incoming links: Construction of small limits from products and equalizers