Every small limit can be constructed as an equalizer between products over the objects and arrows of the index category

By [F3] the two displayed families are set-indexed. By [F1], the stated coordinate equations define unique maps $s,t:P\to Q$.

Put ${\lambda }_j=p_{j}e$. Since $se=te$, equality of the $u$-coordinates says $D(u){\lambda }_j={\lambda }_k$ for every $u:j\to k$. Thus $\lambda$ is a cone.

Given a cone ${\xi }_j:X\to D(j)$, [F1] supplies a unique $a:X\to P$ with $p_{j}a={\xi }_j$. Its cone equations imply $q_{u}sa=D(u){\xi }_j={\xi }_k=q_{u}ta$ for all $u$, so product uniqueness gives $sa=ta$.

By [F2], $a$ factors uniquely as $a=eh$ with $h:X\to L$. Then ${\lambda }_{j}h=p_{j}eh={\xi }_j$, so $h$ is a cone morphism.

If $h^{\mathrm{\prime }}$ has the same leg equations, product uniqueness gives $e{h}^{\mathrm{\prime }}=a=eh$; equalizer uniqueness gives $h^{\mathrm{\prime }}=h$.

Steps 2.1, 2.2, 3.1, and 4.1 are exactly [F4], so $(L,\lambda )$ is a limit. If $\mathcal{J}$ is empty, both products are terminal objects, $s=t$, and their equalizer is isomorphic to the terminal object, so the construction still applies.