I am reading the article "A convenient category for directed homotopy" by Fajstrup and Rosicky and I have a doubt about the proof of Proposition 3.5. The setting is the following:
let $\cal{C}$ be a concrete category, with forgetful functor $U\colon \cal{C}\to \mathsf{Set}$.


**Definition 1.** A full subcategory $\cal{I}$ of $\cal{C}$ is said to be *finally dense* in $\cal{C}$ if, for every object $X$ in $\cal{C}$ there exists a family of maps $(f\_\lambda\colon C\_{\lambda}\to X)\_{\lambda\in \Lambda}$ such that a map $g\colon UX\to UY$ lifts to a morphism $\bar{g}\colon X\to Y$ iff $g\circ Uf\_\lambda$ lifts to a morphism $\bar{g\_\lambda}$ for every $\lambda$.


**Definition 2.** A full subcategory $\cal{I}$ of $\cal{C}$ is said to be *dense* in $\cal{C}$ if every object $X$ in $\cal{C}$ is isomorphic to the colimit of the canonical functor $\mathcal{I}/X\to \cal{C}$.


In Proposition 3.5 of the aforementioned paper, $\cal{C}$ is actually the category $\cal{K}\_{\cal{I}}$ of $\cal{I}$-*generated* objects, for a topological construct $\cal{K}$ and a full subcategory $\cal{I}$ of $\cal{K}$. In particular, this means that we have a forgetful functor $U\colon\cal{K}\to \mathsf{Set}$ which has a right and a left adjoint, that associate to every set $S$ the discrete and indiscrete object on $S$, respectively.
The statement I'm a bit puzzled about is that, "since $\cal{I}$ is finally dense in $\cal{C}$, then it is dense."


I can see why this should be true if we could show that, for any $X\in \cal{K}$ the natural map 
$$\mathrm{colim}\_{C\to X}UC\to UX$$
is a bijection.
If either $\cal{I}$ contains the discrete object on the terminal object, or the discrete object and the indiscrete object on the terminal object conincide, then I know how to conclude, since the aformentioned cocone would contain all the points of $UX$ (In the second case it is enough to consider all constant maps $C\to \*\_I=\*\_D\to X$. However, in a more general setting I don't know why such a statement should hold.


Here comes the question:


Am I missing something? Is there a reason why the above statement should hold in a more general setting?


If the answer is no, do you have any example of a topological category where UX and the canonical colimit displayed above do not coincide?



Finally dense implies dense
You are right, in general $\mathcal I$ should contain the discrete object $D\_1$ on the singleton. A general result is in my paper Codensity and binding categories (Theorem 1.3). In Proposition 3.5 of the aforementioned paper, $D\_1$ is $\mathcal I$-generated and thus it can be added to $\mathcal I$ without changing $\mathcal I$-generated objects. Then Proposition 3.5 follows.


