 Solution

Identify the reflective subcategory with its full replete image, writing ${\eta }_X:X\to LX$ for the reflection unit. An object $Z$ is fixed by $L$ exactly when precomposition with every ${\eta }_X$ gives a bijection $\mathcal{E}(LX,Z)\to \mathcal{E}(X,Z)$. One implication is the reflection adjunction; for the other, take $X=Z$ to obtain a retraction of ${\eta }_Z$, and uniqueness of maps into the fixed object $LZ$ makes it a two-sided inverse.

An ambient finite limit of fixed objects has this same property: its hom-sets are limits of the hom-sets of the diagram, and a limit of these bijections is a bijection. Thus the ambient terminal object, products and equalizers are fixed and retain their universal properties in the full subcategory. The reflective subcategory is Cartesian, with finite limits computed in the ambient topos.