Remark 1.2.1.2. In the situation of Definition 1.2.1.1, if $S'_{\bullet } \subseteq S_{\bullet }$ is a summand, then the complementary summand $S''_{\bullet }$ is uniquely determined: for each $n \geq 0$, we must have $S''_{n} = S_{n} \setminus S'_{n}$. Consequently, the condition that $S'_{\bullet }$ is a summand of $S_{\bullet }$ is equivalent to the condition that the construction
\[ ( [n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto S_{n} \setminus S'_{n} \]
is functorial: that is, that the face and degeneracy operators for the simplicial set $S_{\bullet }$ preserve the subsets $S_{n} \setminus S'_{n}$.