This question is essentially a follow-up to this question. But before entering the question, let me present the relevant definitions contained in *The joy of cats*.

**Definition 1.** Leave $ bf {X} $ be a category ONE *specific category about $ bf {X} $* it's a pair $ ({ bf {A}}, U) $, where $ bf {A} $ it's a category and $ U: { bf {A}} a X $ It is a faithful functor.

**Definition 2** Yes $ ({ bf {A}}, U) $ Y $ ({ bf {B}}, V) $ are specific categories about $ bf {X} $ , Then a *concrete functor*

$ ({ bf {A}}, U) $ Y $ ({ bf {B}}, V) $ it's a functor $ F: { bf {A}} a { bf {B}} $ with $ U = V circ F $. We denote such functor

by $ F: ({ bf {A}}, U) to ({ bf {B}}, V) $.

**Definition 3.** Yes $ ({ bf {A}}, U) $ Y $ ({ bf {B}}, V) $ are specific categories about $ bf {X} $ , then a concrete functor of

$ F: ({ bf {A}}, U) to ({ bf {B}}, V) $ it is said to be a concrete isomorphism if $ F: { bf {A}} a { bf {B}} $ It is an isomorphism.

**Definition 4.** Yes $ ({ bf {A}}, U) $ Y $ ({ bf {B}}, V) $ are specific categories about $ bf {X} $ Y $ E: mathbf {A} hookrightarrow mathbf {B} $ Be the inclusion functor. So $ ( mathbf {A}, U) $ it's called a concrete subcategory of $ ( mathbf {B}, V) $ Yes $ U = V circ E $.

**Definition 5.** Leave $ ({ bf {A}}, U) $ Y $ ({ bf {B}}, V) $ are specific categories about $ bf {X} $ such that $ ({ bf {A}}, U) $ It is a specific subcategory of $ ({ bf {B}}, V) $. So $ ({ bf {A}}, U) $ it is said to be a concretely reflective subcategory of $ ({ bf {B}}, V) $ Yes

(1) for each $ mathbf {B} $-object $ B $ there is $ mathbf {B} $-morphism $ r: B a A $ (where $ A $ is a $ mathbf {A} $-object) such that for any $ mathbf {A} $-object $ A & # 39; $ and any $ mathbf {B} $-morphism $ f: B a A & # 39; $, there is only one $ mathbf {A} $-morphism $ g $ such that $ g circ r = f $. These $ r $called $ mathbf {A} $reflection arrows for $ B $& # 39; s.

(2) for each one $ r $ we have $ V (B) = V (A) $ Y $ V (r) = text {id} _ {V (A)} $.

**Definition 6.** Leave $ ({ bf {A}}, U) $ Y $ ({ bf {B}}, V) $ are specific categories about $ bf {X} $ such that $ ({ bf {A}}, U) $ it is a concrete reflective subcategory of $ ({ bf {B}}, V) $. So he **fuctor reflector** so induced is called a **concrete reflector**.

Yes $ mathbf {A} $ Y $ mathbf {B} $ be two categories and $ mathbf {A} $ It is a subcategory of $ mathbf {B} $, so $ mathbf {A} $ it's called a **reflective subcategory of $ mathbf {B} $** If (1) of the above you are satisfied. In this terminology **Definition 5** just say yes $ ({ bf {A}}, U) $ Y $ ({ bf {B}}, V) $ are specific categories about $ bf {X} $ such that $ ({ bf {A}}, U) $ it is a specifically reflexive subcategory of $ ({ bf {B}}, V) $ then in particular $ mathbf {A} $ it is a reflexive subcategory of $ mathbf {B} $.

**My attempt**

Motivated by this response, I first tried to conceptualize the concrete reflector as a reflector that is a concrete functor. But unfortunately this is not the case. Then I tried to conceptualize the concrete reflector as a reflector that is a concrete functor that also retains the information that " $ mathbf {X} $-objects of the domain and codomain of a $ mathbf {A} $reflection arrow for a $ mathbf {B} $-object $ B $ it's the same. "But frankly, this is just reinterpreting the definition in a different language and, therefore, I'm not satisfied with this and I think there must be some deep reason to add this condition.

While looking for a motivation for the reason for the presentation (2), I came across J. Fiadeiro's book *Software Engineering Categories*. There it is written that,

Intuitively, so that the (co) reflection is

"concrete", that is, to be consistent with the classification that the underlying

Functor provides, we would like to stay within the same fiber. That is to say,

we would like the arrows of (co) reflection to be identities.

But I didn't understand this comment since I don't even have a vague and intuitive idea for a specific functor (the answers to the question I linked above only focus on the point of view of the concrete notion *isomorphism*)

**Question**

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