ScalingStacks

We now prove that ([00BZ]) induces an equivalence on the hom-space between any pair of objects \(\{\mathcal C_1 \rightarrow\mathcal D_1\}, \{\mathcal C_2 \rightarrow\mathcal D_2\} \in\ \textrm{Ar}^{(n-1)}(\mathrm{Cat}_{(\infty, {k})})\), and hence that ([00BV]) is fully faithful. Unwinding the hom-spaces in the relevant arrow \(\infty\)-categories, this is equivalent to the statement that for any fixed \(\mathcal D_1 \rightarrow\mathcal D_2\) and any fixed dashed lift as shown in the first diagram in ([00C0]), the space of dashed lifts as shown in the commuting square in the second diagram in ([00C0]) is contractible. By lemma 5.3.14, \(\iota_n \mathcal C_1 \rightarrow\mathcal C_1\) is \((n-1)\)-surjective, and \(\mathcal C_2 \rightarrow\mathcal D_2\) is \((n-1)\)-faithful by assumption, hence the space of lift is contractible since \((n-1)\)-surjective/\((n-1)\)-faithful functors form a factorization system on \(\mathrm{Cat}_{(\infty, {k})}\). ◻