= 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 $LA=A$,
$$
L(X\times A)\cong LX\times A,
$$
and its unit corresponds to $\eta_X\times1_A$. Because $B$ is fixed, every map $X\times A\to B$ therefore extends uniquely to $LX\times A\to B$. Transposing back gives
$$
\mathcal E(LX,B^A)\cong\mathcal E(X,B^A).
$$
The 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 \b[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.
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