 Solution

Let $A,B$ be fixed objects and form the ambient exponential object $B^A$. For every $X$, the exponential adjunction identifies maps $X\to B^A$ with maps $X\times A\to B$. Since $L$ is left exact and $LA=A$,and its unit corresponds to ${\eta }_X\times 1_A$. Because $B$ is fixed, every map $X\times A\to B$ therefore extends uniquely to $LX\times A\to B$. Transposing back givesThe fixed-object criterion from part (i) shows that $B^A$ already belongs to $\mathcal{L}$. The ambient evaluation and transpose universal property restrict to the full subcategory, proving the subcategory is cartesian closed. This constructs exponentials in a left-exact reflective subcategory, rather than merely reflecting an exponential and assuming its universal property survives.