For a set \(X\), we denote by \(UX\) the set of ultrafilters on \(X\). For a map \(f:X \rightarrow Y\), the map \(Uf: UX \rightarrow UY\) sends \(x\in UX\) to its image under \(f\), defined by \[B\in Uf(x)\Leftrightarrow f^{-1}(B)\in x,\] for all \(B\subseteq Y\).
Definition 2.11. Denoting its reflection into \(U\)-ASpa by \(\beta_{X}:X\rightarrow BX\) (with \(B\) to be read as capital Beta), we say that a \(U\)-space \(X\) satisfies condition

(C)

if \(X\) carries the cartesian (= initial = weak) structure with respect to \(\beta_{X}\); that is, if for all \(x\in UX, y\in X\), one has \(x\rightsquigarrow y\) in \(X\) whenever \(U\beta_{X}(x)\rightsquigarrow \beta_{X}y\) in \(BX\);

(F)

if \(\beta_{X}\) is a monomorphism; that is: if \(\beta_{X}\) an injective map;

(CF)

if \(X\) satisfies (C) and (F).

In this paper the following results are proved:
Theorem 2.10. The category \(U\)-ASpa is reflective in \(U\)-RGph and, hence, also in \(U\)-Spa.
Theorem 2.12. Consider (a possible large or empty) family of monotone maps \(f_{i}:X \rightarrow Y_{i}(i\in I)\) of \(U\)-spaces with the given common domain \(X\).

(1)

Let \(X\) carry the weak (= initial) structure with respect to \((f_{i})_{i\in I}\). Then, if every \(Y_{i}\) satisfies (C), so does \(X\).

(2)

Let \((f_{i})_{i\in I}\) be point-separarating. Then, if every \(Y_{i}\) satisfies (F), so does \(X\).

(3)

\(U\)-CSpa is simultaneously epi- and mono-reflective in \(U\)-Spa, and \(U\)-FSpa is (regular epi)- reflective in \(U\)-Spa.

Theorem 4.7. \(T\)-\(U\)-ASpa(C) is reflective in \(T\)-\(U\)-Spa(C).
Theorem 4.11. Let \(f_{i} : X \rightarrow Y_{i} (i \in I)\) be (a possibly large) family of monotone morphisms of \(T\)-spaces with common domain \(X\). Then:

(1)

If \(X\) carries the initial structure respect to \((f_{i})_{i\in I}\) and the topological functor \(t\)-Spa(C)\(\rightarrow C\), and if every \(Y_{i}\) satisfies (C), then so does \(X\).

(2)

If \((f_{i})_{i\in I}\) is collectively monic, and if every \(Y_{i}\), satisfies (F), then so does \(X\).

(3)

\(T\)-CSpa(C) is simultancously epi-and mono-reflective in \(T\)-Spa(C).

(4)

\(T\)-FSpa(C) is (regular epi)-reflective in \(T\)-Spa(C).

(5)

\(T\)-CFSpa(C) is epi-reflective in \(T\)-Spa(C).