Let $(\mathscr{A},U)$ be a topological construct (that is, a concrete category $U\colon\mathscr{A}\to Set$ where every structured sink has a unique final lift) and let $\mathscr{B}\subset\mathscr{A}$ be a full subconstruct (say concretely reflective, if it helps).
Given an object $X\in\mathscr{B}$ and a map $f\colon UX \to S$ onto a set S, we can lift $f$ to a quotient morphism either in $\mathscr{A}$ or in $\mathscr{B}$.
Clearly the two lifts are different in general, but:
- Can we say at least that the codomain of either lift always lies in $\mathscr{B}$? (So the question is really whether the codomain of the $\mathscr{A}$-lift lies in $\mathscr{B}$).
- If not, are there conditions on $\mathscr{A}$ and $\mathscr{B}$ ensuring an affirmative answer?
EDIT: The quotient of a $T_0$ space is not $T_0$ in general, so this provides a negative answer to question 1 ($T_0$ spaces form a full reflective subconstruct of $Top$). Question 2 is still open, any idea?
EDIT2: Every topological property preserved by quotient maps selects a subcategory of $Top$ that gives a positive example for the problem above. For instance, the subcategory of compact spaces, of connected spaces, of path-connected spaces, of discrete spaces, all are positive examples. I don't see a common abstract property for these subconstructs though.