= Solution
Let $J:\mathcal C\hookrightarrow\mathcal B$ exhibit a <reflective subcategory>, with <reflector> $K$, and let $E:I\to\mathcal C$ be a small <diagram in a category>. Completeness of $\mathcal B$ gives a <categorical limit> $L$ of $JE$. For every $B'\in\mathcal B$, the <universal property> of this <categorical limit> and the reflection <adjunction> give
$$
\begin{aligned}
\mathcal B(JKB',L)
&\cong\lim_i\mathcal B(JKB',JE_i)\\
&\cong\lim_i\mathcal B(B',JE_i)\\
&\cong\mathcal B(B',L).
\end{aligned}
$$
The composite is precomposition with $\rho_{B'}$, so $L$ satisfies the <hom-set> condition from the preceding part. Its proof of invertibility of $\rho_L$ did not require repleteness. Thus $L\cong JKL$ whether or not the chosen full reflective subcategory is replete.
Transport the ambient limit cone along $\rho_L^{-1}:JKL\to L$. Its legs lie in the <full subcategory>, and their ambient <universal property>, restricted to objects of $\mathcal C$, is exactly the internal <categorical limit> property. Hence \b[every small diagram in the reflective subcategory has a limit]:
$$
\boxed{\mathcal C\text{ is complete, with a chosen limit }KL.}
$$
For a <replete subcategory>, the ambient limit object $L$ itself belongs to $\mathcal C$. This argument includes the empty diagram and requires no limit-preservation hypothesis on the <reflector>.
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