Definition 8.6.6.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, so that the induced map
\[ \lambda : \operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}} \]
is a left fibration (Proposition 8.1.1.15). Let $\kappa $ be an uncountable cardinal. We say that a diagram
\[ \mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa } \]
is a relative $\operatorname{Hom}$-functor for $U$ if it is a covariant transport representation for the left fibration $\lambda $ (Definition 5.6.5.1).