A reflective inclusion creates every ambient limit in the ordinary isomorphism-invariant sense

Applying $R$ to the legs and then the counit gives a cone from $R(L)$ to $D$; after inclusion its legs are $I({\epsilon }_{D_j})\circ IR(p_j)$. Naturality of the unit and the triangle identity give $I({\epsilon }_{D_j})\circ IR(p_j)\circ {\eta }_L=p_j$, so ${\eta }_L$ is a morphism from the given cone to this included cone.

By the universal property in [L4], the included cone has a unique map $q:IR(L)\to L$ with $p_j\circ q=I({\epsilon }_{D_j})\circ IR(p_j)$. Cone uniqueness gives $q\circ {\eta }_L=1_L$. Both $1_{IR(L)}$ and ${\eta }_L\circ q$ are maps between reflected objects whose composites with the universal reflection arrow ${\eta }_L$ equal ${\eta }_L$, and by [L5] the unit component ${\eta }_L$ is a universal arrow from $L$ to $I$, so the uniqueness clause of that universal property makes ${\eta }_L\circ q=1_{IR(L)}$. Hence ${\eta }_L$ is invertible and [L1] identifies $L$ with an included object.

Transporting the limiting cone across this isomorphism produces a cone of $D$ in the full subcategory whose image is isomorphic to $(L,p_j)$. Its image is limiting, and since $I$ is a right adjoint, [L2] preserves every source limit; conversely fullness makes any mediating map between included objects a unique map in $\mathcal{A}$, so a source cone with limiting image is limiting. These are exactly the clauses of [L3], including the empty and degenerate indexing categories.