All the definitions that follow is taken from The Joy of Cats.

Definition 1.Let $\bf{X}$ be a category. Aconcrete 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 aconcrete 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.

## Question

We know that isomorphic categories can be viewed as being "essentially same". But then how should I view concretely isomorphic concrete categories? More generally how should I view a concrete functor?

In The Joy of Cats the following is written,

A concrete isomorphism $F : (\mathbf{A},U) \to (\mathbf{B}, V )$ between concrete categories over $\bf{X}$ is a concrete functor that is an isomorphism of categories. ...That such a concrete isomorphism exists means, informally, that each structure in $\bf{A}$, i.e., each object $A$ of $\bf{A}$, can be completely substituted by a structure in $\bf{B}$, namely $F(A)$ (keeping, of course, the same morphisms). For example, the standard descriptions of topological spaces by means of

• neighborhoods,

• open sets,

• closure operators, or

• convergent filters,

give technically different constructs, all of which are concretely isomorphic. This is why the differences between the various descriptions are regarded as inessential and we can in good conscience call each of them "$\bf{Top}$". The concept of concretely isomorphic concrete categories gives rise to an equivalence relation that is stronger than the relation of isomorphism of categories. For example, assuming that no measurable cardinals exist, $\bf{Top}$ (and, indeed, any construct) can be thought of as being isomorphic to a full subcategory of $\bf{Rel}$. However, $\bf{Top}$ is not concretely isomorphic to such a subcategory, because there are more topologies on $\Bbb{N}$ (namely, $2^{2^{\aleph_0}}$) than there are binary relations on $\Bbb{N}$ (namely, $2^{\aleph_0}$).

Here $\bf{Top}$ denotes the category of topologies and continuous functions and $\bf{Rel}$ denotes the category whose objects are pairs of the form $(X,\rho)$ (where $\rho$ is a binary relation on the set $X$) and whose morphisms are relation-preserving functions.

But I don't understand why the same description as given in the following, "That such a concrete isomorphism exists means, ...the same morphisms)." can't be said about isomorphic categories.