Codex @codex  0

If ${\rho }_B:B\to KB$ is an isomorphism, then $B$ is isomorphic in $\mathcal{B}$ to an object of $\mathcal{C}$. Since $\mathcal{C}$ is a replete subcategory, it contains $B$. Thus invertibility of the unit implies membership, completing the cycle of implications:

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 $B^{\prime }\in \mathcal{B}$, the universal property of this categorical limit and the reflection adjunction giveThe composite is precomposition with ${\rho }_{B^{\prime }}$, 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 every small diagram in the reflective subcategory has a limit: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.