Here's the question I think I'm asking, with background below if necessary:
Question: The reflector $L$ left adjoint to the inclusion of a reflective subcategory $\mathcal L\to\mathcal E$ is characterized by the fact that if $A\in\mathcal{L}$ every map $B\to A$ factors uniquely through $LB$ via the unit $\eta_B$. Is the converse-namely that if every map $B\to A$ factors uniquely through the reflection $LB$, then $A\in\mathcal{L}$-true? Perhaps just if we require $\mathcal E$ to be cartesian closed?
Background: It's (part of) proposition A4.3.1 in Sketches of an Elephant that
Let $\mathcal E$ be a cartesian closed category and $\mathcal L$ a reflective subcategory of $\mathcal E$ with reflector $L$. Then $L$ preserves finite products iff the class of objects of $\mathcal L$ is an exponential ideal.
Let me recall that $L$ is a left adjoint of the inclusion $\mathcal L\to \mathcal E$ and that $\mathcal L$ is an exponential ideal if $A^B$ is in $\mathcal L$ whenever $A\in\mathcal L$ and $B\in\mathcal E$.
In proving that finite product-preservation implies $\mathcal L$ is an exponential ideal, Johnstone goes through a chain of adjunctions to show that each morphism $C\to B^A$ with $B\in\mathcal L$ factors uniquely through the unit of the adjunciton $\eta_C:C\to LC$. He then claims
...applying this to the identity morphism $B^A\to B^A$, we deduce that $\eta_{B^A}$ is an isomorphism.
This looks like a throwaway comment, but I'm having trouble verifying it. Looking at the identity of $B^A$ does show that $\eta_{B^A}$ is a split monomorphism. But it's certainly not true in general that if the identity on an object $X$ factors uniquely through a split monomorphism $m$ out of $X$ that $m$ is an isomorphism. So I took the splitting $h$ of $\eta_{B^A}$ and tried to show it's monic. I can get that $hf=hg$ implies $hLf=hLg$, but that's no use. I haven't been able to make any real use of cartesian closedness.
We are given that $\eta_{B^A}$ is a split mono with $h\circ\eta_{B^A}=1_{B^A}$. By naturality of the unit $\eta_{B^A}\circ h=Lh\circ \eta_{LB^A}$ with $\eta_{LB^A}$ iso (the unit is iso on its subcategory). But by the triangle identities for the adjunction, $Lh=\epsilon_{LB^A}$. The inclusion functor is faithful by definition, hence the counit $\epsilon$ is epi. Therefore $Lh$ is epi implying $\eta_{B^A}$ is epi as well as split mono, in other words, iso.