Notation 8.3.3.7. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. We will often use the notation $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, - )$ to denote a $\operatorname{Hom}$-functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$. Beware that this convention introduces a slight potential for confusion. Given a pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have two potentially different definitions of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$:
- $(a)$
The Kan complex $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $ of Construction 4.6.1.1, which is well-defined up to canonical isomorphism.
- $(b)$
The Kan complex $\mathscr {H}(X,Y)$, which is only well-defined up to homotopy equivalence (since it depends on a choice of $\operatorname{Hom}$-functor $\mathscr {H}$).
However, the danger is slight: Remark 8.3.3.3 guarantees the existence of homotopy equivalences $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \simeq \mathscr {H}(X,Y)$, which can be chosen to depend functorially on $X$ and $Y$ (as morphisms in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$). Consequently, we can always modify the choice of $\operatorname{Hom}$-functor $\mathscr {H}$ to arrange that definitions $(a)$ and $(b)$ coincide (see Corollary 4.4.5.3).