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).