Joy of Cats Corollary 10.50 I have been studying category theory from Joy of Cats. I am stuck at proving Corollary 10.50 from Proposition 10.49. Which says that,

Embeddings of concretely reflective subcategories preserve initial sources.

Since I was trying to apply Proposition 10.49 I was thinking about somehow showing that the concrete reflector (say $R$) and the embedding (say $E$) forms a Galois correspondence. Then by Proposition 10.49 we could conclude that $E$ preserves initial sources. But that's precisely where I am stuck. Can anyone help?

For the sake of completeness here are 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. Let $(\mathbf{A},U)$ and $(\mathbf{B}, V)$ are concrete categories over $\mathbf{X}$ and $R:(\mathbf{B},V)\to (\mathbf{A}, U)$ is a concrete functor. Then we call $F$ to be a concrete reflector if the following holds:

*

*for all $\mathbf{B}$-object $C$ there exists a $\mathbf{B}$-morphism $r_C:C\to R(C)$ such that $V(r_C)=id_{V(C)}$.

*for all $\mathbf{B}$-objects $B,B'$ and a $\mathbf{B}$-morphsim $f:B\to B'$, $R(f)$ is the unique $\mathbf{A}$-morphism making the following diagram, $$\require{AMScd}
\begin{CD}
B @>{r_B}>> R(B);\\
@V{f}VV @VV{R(f)}V \\
B' @>>{r_{B'}}> R(B');
\end{CD}$$commute.

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 $id_{\mathbf{B}}\le G\circ F$.
Definition 6. Let $(\mathbf{A},U)$ be a concrete category A family of $\mathbf{A}$-morphisms $(A\overset{f_i}{\longrightarrow}A_i)_{i\in I}$ is said to be an initial source if for any family of $\mathbf{A}$-morphisms $(B\overset{k_i}{\longrightarrow}A_i)_{i\in I}$ and for any $\mathbf{X}$-morphism $U(B)\overset{h}{\longrightarrow}U(A)$, $$U(B\overset{g}{\longrightarrow}A)=U(B)\overset{h}{\longrightarrow}U(A)$$whenever $U(k_i)=U(f_i)\circ h$ for all $i\in I$.

 A: Let me first say that when it says embeddings of concretely reflective subcategories preserve initial sources, it means that if $A$ is a concrete subcategory of $B$ over $X$, and $E$ is the embedding $A\newcommand\into\hookrightarrow\into B$, and moreover $A$ admits a concrete reflection $R$, then $E$ preserves initial sources.
If you could prove that any embedding of $A$ into any other category preserved initial sources, just because it was a concretely reflective subcategory of some particular category $B$, then in particular, we can always realize $A$ as a concretely reflective subcategory of itself, so this would imply that all embeddings preserve initial sources. This is false, and I'm pretty sure you can come up with nice finite categories to give a counterexample.
With that said, then it should suffice to prove 6.26, and you've asked about it, so I'll give an explanation here.

Example 6.26 (2) We want to show that if $E : A\to B$ is a concrete embedding, and
$R:B\to A$ is a concrete reflector, then $(R,E)$ is a Galois correspondence.

Proof.
Let's be careful about the definition of concrete reflector. In particular, saying that we have a $B$-morphism $r_b:b\to Rb$ is actually slightly abusive of notation. $Rb$ belongs to $A$, so when we say $r_b:b\to Rb$, we actually mean $r_b:b\to ERb$. Also the requirement that $|r_b| = 1_{|b|}$ tells us that $\mathrm{id}_B\le ER$. Thus we just need to show that $RE\le \mathrm{id}_A$.
To show this, we need to use one more piece of information, that the arrows $r_b$ are $A$-reflection arrows (see definition 4.16). This implies in particular, that for all $A$-objects $a$, there is a unique $A$-morphism $s_a : REa\to a$ such that the following diagram commutes (taking great care to use the embedding $E$ explicitly):
$$
\require{AMScd}
\begin{CD}
Ea @>r_{Ea}>> EREa\\
@| @VVE s_aV
\\
Ea @>1_{Ea}>> Ea \\
\end{CD}
$$
Now when we take the underlying $X$-objects of this diagram, we get
$$
\begin{CD}
|a| @>1_{|a|}>> |a|\\
@| @VV |s_a|V
\\
|a| @>1_{|a|}>> |a|, \\
\end{CD}
$$
so $|s_a| = 1_{|a|}$. Thus the morphisms $s$ show $RE\le \textrm{id}_A$, as desired.
