Before going into the main problem at which I am stuck, let me give first the relevant definitions.
Definition 1. Let $\mathbf{X}$ be a category. A concrete category over $\mathbf{X}$ is a pair $(\mathbf{A},U)$, where $\mathbf{A}$ is a category and $U : \mathbf{A} \to \mathbf{X}$ is a faithful functor.
Definition 2. If $(\mathbf{A},U)$ and $(\mathbf{B}, V)$ are concrete categories over $\mathbf{X}$ , then a concrete functor from $(\mathbf{A},U)$ to $(\mathbf{B}, V)$ is a functor $F : \mathbf{A}\to \mathbf{B}$ with $U = V \circ F$. We denote such a functor by $F : (\mathbf{A},U)\to (\mathbf{B},V)$.
Definition 3. If $(\mathbf{A},U)$ is a concrete category over $\mathbf{X}$ then we say that $(\mathbf{A},U)$ is amnestic iff $U$ is an amnestic functor (i.e, if for some $\mathbf{A}$-isomorphism $f$, $U(f)$ is a $\mathbf{X}$-identity, then $f$ must also be an $\mathbf{A}$-identity).
Definition 4. Let $(\mathbf{A},U)$ and $(\mathbf{B}, V)$ are concrete categories over $\mathbf{X}$ and $F,G:(\mathbf{A},U)\to (\mathbf{B}, V)$ be two concrete functors. Then we write $F\le G$ iff for all $\mathbf{A}$-object $A$ there exists a $\mathbf{B}$-morphism $f:F(A)\to G(A)$ such that $V(f)$ is a $\mathbf{X}$-identity.
Definition 5. Let $(\mathbf{A},U)$ and $(\mathbf{B}, V)$ are concrete categories over $\mathbf{X}$ and $F:(\mathbf{B},V)\to (\mathbf{A}, U)$ and $G:(\mathbf{A},U)\to (\mathbf{B},V)$ be two concrete functors. Then we say that $(F,G)$ is a Galois correspondence if $F\circ G\le id_{\mathbf{A}}$ and $id_{\mathbf{B}}\le G\circ F$.
Now the problem with which I am stuck is regarding Corollary 6.32 of Joy of Cats which is stated as follows (I am quoting only the relevant parts),
Corollary 6.32. Let $G : \mathbf{A}\to \mathbf{B}$ and $F : \mathbf{B}\to\mathbf{A}$ be concrete functors between amnestic concrete categories such that $(F,G)$ is a Galois correspondence, ...and $\mathbf{B}^\ast$ be the full subcategory of $\mathbf{B}$ with objects: $\{G(A) |A \in Ob(\mathbf{A})\}$... . Then $\mathbf{B}^\ast$ is reflective in $\mathbf{B}$, and $\mathbf{B}\in Ob(\mathbf{B}^\ast)$ if and only if $B = (G \circ F)(B)$.
I have managed to prove that,
Let $G : \mathbf{A}\to \mathbf{B}$ and $F : \mathbf{B}\to\mathbf{A}$ be concrete functors between amnestic concrete categories such that $(F,G)$ is a Galois correspondence, and let $\mathbf{B}^\ast$ be the full subcategory of $\mathbf{B}$ with objects: $\{G(A) |A \in Ob(\mathbf{A})\}$... . Then the followings hold,
- If $B\in Ob(\mathbf{B}^\ast)$ then $B = (G \circ F)(B)$.
- $\mathbf{B}^\ast$ is reflective in $\mathbf{B}$.
What we remains to be shown is that if $B = (G \circ F)(B)$ then $B\in Ob(\mathbf{B}^\ast)$. But I can't prove this. Can anyone give me any idea regarding how to prove it?
If an object $B$ of $\boldsymbol{B}$ is such that $B=(G\circ F)(B)=G(FB)$, then in particular $B$ is an object of $\boldsymbol{B}^*$, since $FB$ is an obect of $\boldsymbol{A}$.