= 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. \b[The reflective subcategory is Cartesian, with finite limits computed in the ambient topos.]
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