Example 6.2.5.2. In the situation of Definition 6.2.5.1, assume that $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category. Then an $S$-optimal factorization of a morphism $f: X \rightarrow Z$ is a pair of morphisms $X \xrightarrow {g} Y \xrightarrow {s} Z$, where $s \in S$ and $s \circ g = f$, which has the following universal property: for every other pair of morphisms $X \xrightarrow {g'} Y' \xrightarrow {s'} Z$ with $s' \in S$ and $s' \circ g' = f$, there is a unique morphism $h: Y \rightarrow Y'$ satisfying $h \circ g = g'$ and $s' \circ h = s$, as indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar@ {-->}[dd]^{h} \ar [dr]^{s} & \\ X \ar [ur]^{g} \ar [dr]_{g'} & & Z \\ & Y'. \ar [ur]_{s'} & } \]
Stated more informally, the pair $(g,s)$ is universal among all factorizations of $f$ through a morphism which belongs to $S$.