Kerodon

In the left corner of the diagram, I wonder if we should use $\operatorname{h}\operatorname{Pith}(\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}))$. I don't know whether or not $\operatorname{Pith}$ preserves the homotopy category of $(\infty,2)$-categories, because we have more 2-simplices in the origional category. In particular, in 5.4.5.12, is this definition of isomorphisms in an $(\infty,2)$-category the same as isomorphisms in its homotopy category?

Yes, the lower left hand corner should be the pith. If I'm understanding your question correctly, the answer is no: given an $(\infty,2)$-category, one can form an $\infty$-category either by discarding all of the noninvertible $2$-morphisms (which is the pith) or by forcing all the noninvertible $2$-morphisms to become invertible. These will generally be different $\infty$-categories which have different homotopy categories, and the lower left hand corner should involve the first of these (in this situation, the noninvertible $2$-morphisms induce non-invertible natural transformations between the transport functors).

Little typos: 1. In the webpage, the brackets in $\mathrm{hPith}(\mathrm{N^{hc}_\bullet}(\mathcal{C}))$ seem not to be depicted as expectation (at least for me). 2. In the diagram of (a), it seems that one shouldn't take core at the left corner. 3. "...$Fg_\sigma$ is an $n$-simplex of...", it seems that $Fg_\sigma$ should be corrected as $g_\sigma$. 4. In the definition of $u_{i,j}$, the target is a simplicial set hence it should be corrected as $\mathrm{Hom}_{\mathcal{C}}(h_{\sigma}(i),h_{\sigma}(j))_{\bullet}$. 5. The enumerations $(a')$, $(b')$, $(c')$ in the definition of $u_{i,j}$ aren't compiled well on the webpage (at least for me).

Moreover, in (02S0) and in the definition of $\lambda$, is the abbreviation $\mathrm{N}^{\rm hc}_\bullet$ for $\mathrm{N}^{\rm hc}_\bullet(\mathcal{C})$ intended or just a typo?

Yep. Thanks!