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**
--------------


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 adding this condition.


While searching for a motivation for the reason for introducing (2), I stumbled upon J. Fiadeiro's book *Categories for Software Engineering*. There it is written that,



> 
> Intuitively, for the (co)reflection to be
>  "concrete", i.e. to be consistent with the classification that the underlying
>  functor provides, we would like to remain within the same fibre. That is,
>  we would like that the (co)reflection arrows be identities.
> 
> 
> 


But I didn't understand this comment since I don't even have a vague intuitive idea for a concrete functor (the answers to the question that I linked above only focuses on the big-picture viewpoint of the notion of concrete *isomorphism*). 


**Question**
------------


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



Understanding the reason for the particular formulation of the definition of a concrete reflector (as stated in The Joy of Cats)
I suppose that there are possibly many different answer to this question. Here is the one I got.


Being a reflector is equivalent to being an inclusion that has a left adjoint. 


In general being a concrete inclusion, thus an inclusion of categories, that is reflective ensure that the inclusion, as a functor of categories, has an adjoint, but there is no garantee that such adjoint is also a concrete functor.


The condition (2) ensures this requirement: a concrete reflector has a left adjoint that is a concrete functor.


This basically grants that concrete reflector are the real reflectors in the 2-categories of concrete categories (over a given category $\mathbf X$): i.e. those inclusion of concrete categories that have a concrete left adjoint.


Hope this helps.


