= Solution
Conversely, assume that every <hom-set> <functor> $\mathcal C(C,F-)$ preserves existing <categorical limits>. For a <categorical limit> cone $\ell_i:L\to E_i$ in $\mathcal B$, a <categorical cone> $q_i:C\to FE_i$ determines an element of $\lim_i\mathcal C(C,FE_i)$. The assumed <bijection>
$$
\mathcal C(C,FL)\longrightarrow\lim_i\mathcal C(C,FE_i),
\qquad u\longmapsto(F\ell_i\,u)_i
$$
gives one and only one such $u$ for every $C$ and every <categorical cone>. That is the <universal property> of the image cone itself. Therefore \b[the hom-set tests detect limit preservation]:
$$
\boxed{F\text{ preserves limits}\iff\mathcal C(C,F-)\text{ does so for every }C.}
$$
This proof works for all diagram sizes for which the relevant <categorical limits> and compatible-family <sets> are under consideration.
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