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If a full subcategory is reflective in a complete category, it is complete. For an ambient limit $L$ of a diagram of reflected objects, the reflection unit $L\to JKL$ is invertible by the hom-set characterization of reflected objects. Thus $KL$ is an internal categorical limit. If the subcategory is replete, $L$ itself belongs to it. The reflector need not preserve arbitrary limits.

If $B$ belongs to the full subcategory $\mathcal{C}$, the adjunction and fullness of its inclusion give, for every $B^{\prime }\in \mathcal{B}$,Therefore membership implies the hom-set condition. In addition, $J$ is fully faithful, so the adjunction counit is invertible by the preceding criterion. The triangular equation $J{\epsilon }_B{\rho }_B=1_B$ then shows that ${\rho }_B$ is an isomorphism. As in the printed inclusion notation, $J$ is suppressed when writing $KB$ as an object of $\mathcal{B}$.

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.