= Solution
Use the explicit <construction of small limits from products and equalizers> from part (i), here with finite indexing <sets>. If $F$ preserves finite <products in a category> and <equalizers>, then $F(P)$ and $F(Q)$, with their image projections, are the products of the objects $FD(j)$ and of the target objects indexed by arrows of $J$.
The image maps $F(a),F(b)$ have components
$$
F(\pi_u)F(a)=FD(u)F(\pi_i),\qquad
F(\pi_u)F(b)=F(\pi_j).
$$
Also $F(e):F(L)\to F(P)$ is their <equalizer>. Therefore the same <universal property> proof makes $F(L)$, with projections $F(p_j)$, a <categorical limit> of $FD$. Any other chosen limit of $D$ is uniquely <isomorphic> to this construction, and the <functor> carries that isomorphism to an isomorphism. \b[Consequently $F$ preserves every finite limit.] Preservation of finite products includes the empty product, so the <terminal object> is also preserved.
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