Definition 5.7.
A morphism \(f\colon X \to Y\) in \(T\) is called an epimorphism if it is \((-1)\)-cotruncated, i.e. if the square
is a pushout square. Let \(T^+\subseteq T\) be the full subcategory of objects \(Z\) for which \(Z\to *\) is right orthogonal to all epimorphisms. The factorization system of Lemma 5.6 gives a reflection \((-)^+\colon T\to T^+\) and hence a factorization
\[X \to X^+ \to *\]
into an epimorphism followed by a map which is right orthogonal to the epimorphisms.