This question is essentially a followup of [this question](https://mathoverflow.net/questions/337370/how-should-i-think-about-concrete-functors-and-in-particular-about-concrete-isom). But before going into the question  let me introduce the relevant definitions as given in [*The Joy of Cats*](http://katmat.math.uni-bremen.de/acc/acc.pdf).

>**Definition 1.** Let $\bf{X}$ be a category. A *concrete category over $\bf{X}$* is a pair $({\bf{A}},U)$, where $\bf{A}$ is a category and $U :{\bf{A}} \to X$ is a faithful functor. 

>**Definition 2.** If $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ , then a *concrete functor from
$({\bf{A}},U)$ and $({\bf{B}}, V )$* is a functor $F : {\bf{A}}\to {\bf{B}}$ with $U = V \circ F$. We denote such a functor
by $F : ({\bf{A}},U)\to ({\bf{B}}, V )$.

>**Definition 3.** If $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ , then a concrete functor from
$F:({\bf{A}},U)\to ({\bf{B}}, V )$ is said to be a concrete isomorphism iff $F:{\bf{A}}\to{\bf{B}}$ is an isomorphism.

>**Definition 4.** If $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ and $E:\mathbf{A}\hookrightarrow\mathbf{B}$ be the inclusion functor. Then $(\mathbf{A},U)$ is called a concrete subcategory of $(\mathbf{B},V)$ if $U=V\circ E$.

>**Definition 5.** Let $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ such that $({\bf{A}},U)$ is a concrete subcategory of $({\bf{B}},V)$. Then $({\bf{A}},U)$ is said to be a concretely reflective subcategory of $({\bf{B}},V)$ if 

>(1) for each $\mathbf{B}$-object $B$ there exists a $\mathbf{B}$-morphism $r:B\to A$ (where $A$ is an $\mathbf{A}$-object) such that for any $\mathbf{A}$-object $A'$ and any $\mathbf{B}$-morphism $f:B\to A'$, there exists an unique $\mathbf{A}$-morphism $g$ such that $g\circ r=f$. These $r$'s are called $\mathbf{A}$-reflection arrows for $B$'s.

>(2) for each such $r$ we have $V(B)=V(A)$ and $V(r)=\text{id}_{V(A)}$.

>**Definition 6.** Let $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ such that $({\bf{A}},U)$ is a concrete reflective subcategory of $({\bf{B}},V)$. Then the **reflector fuctor** thus induced is called a **concrete reflector**. 


If $\mathbf{A}$ and $\mathbf{B}$ be two categories and $\mathbf{A}$ is a subcategory of $\mathbf{B}$, then $\mathbf{A}$ is called a **reflective subcategory of $\mathbf{B}$** if (1) of the above is satisfied. In this terminology **Definition 5** simply says that if $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ such that $({\bf{A}},U)$ is a concretely reflective subcategory of $({\bf{B}},V)$ then in particular $\mathbf{A}$ is a reflective subcategory of $\mathbf{B}$. 

**My Attempt**
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Motivated by [this answer](https://mathoverflow.net/a/337380/57432), I first tried to conceptualize the concrete reflector as a reflector which is a concrete functor. But unfortunately this is not the case. Then I tried to conceptualize the concrete reflector as a reflector which is a concrete functor that also preserves the information that "the underlying $\mathbf{X}$-objects of the domain and codomain of a $\mathbf{A}$-reflection arrow for a $\mathbf{B}$-object $B$ is same". But frankly, this is just reinterpreting the definition in a different language and hence I am not satisfied with this and think that there must be some deep reason for doing this. 

**Question**
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I am trying to understand the reason for adding (2). What is the motivation for this?