= Solution
Assume $F$ preserves existing <categorical limits>. For a <diagram in a category> $E:I\to\mathcal B$ with <categorical limit> cone $\ell_i:L\to E_i$, its image $F\ell_i$ is a <categorical limit> cone of $FE$. For any $C\in\mathcal C$, a <morphism> $C\to FL$ is therefore uniquely equivalent to a compatible family of <morphisms> $C\to FE_i$. In the <Category of sets>, compatible families are exactly the <categorical limit> of the resulting <hom-sets>. Thus
$$
\boxed{\mathcal C(C,FL)\cong\lim_i\mathcal C(C,FE_i).}
$$
The maps are induced by $F\ell_i$, so \b[$\mathcal C(C,F-)$ preserves the given limit]. Applying this to every existing <categorical limit> proves the implication, with no completeness hypothesis on $\mathcal B$.
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