= Solution
Let $D:I\to\mathcal C$ be a finite diagram. Form the finite <categorical products>
$$
P=\prod_{i\in\operatorname{Ob}I}D(i),\qquad Q=\prod_{u:i\to j\in\operatorname{Mor}I}D(j).
$$
There are two <morphisms> $s,t:P\rightrightarrows Q$: their components at $u:i\to j$ are $D(u)\pi_i$ and $\pi_j$, respectively. Let $e:L\to P$ be their <equaliser>. Then $\ell_i=\pi_i e$ satisfy $D(u)\ell_i=\ell_j$, so they constitute a <cone over a diagram>.
Any other cone $x_i:X\to D(i)$ uniquely determines $x:X\to P$. Its cone equations say precisely that $sx=tx$. The <equaliser> therefore gives a unique $h:X\to L$ with $eh=x$, and thus $\ell_i h=x_i$. Conversely these component equations force $eh=x$ by the <categorical product> <universal property>, so there is exactly one such mediator. This proves that $(L,\ell_i)$ is a <categorical limit> of $D$.
The empty <categorical products> are included: if $I$ is empty, $P$ and $Q$ are terminal and the construction gives the empty <categorical limit>. Thus \b[finite <categorical products> and <equalisers> give every finite <categorical limit>], as a special case of the <construction of small limits from products and equalizers>.
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