Remark 9.3.3.2. In the situation of Definition 9.3.3.1, the composition map
\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \xrightarrow { [f] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y) \]
is only well-defined up to homotopy (see Notation 4.6.9.15). However, the condition that $\theta $ is $n$-truncated depends only on its homotopy class (Remark 3.5.9.5).