As noted in [Tag 01WZ of Kerodon](https://kerodon.net/tag/01WZ), the morphism spaces of an $(\infty,2)$-category $C$ can fail to be $\infty$-categories, [in contrast to the _pinched_ morphism spaces of $C$](https://kerodon.net/tag/01WY).

This feels very counterintuitive to me, as the only difference between the former and the latter is that morphism spaces have in a sense "two times as many homotopies" than the pinched morphism spaces: for instance, the 1-simplices of $Hom(A,B)$ look like this:

<img src="https://i.sstatic.net/oPuab.png" width="300"/>

i.e., a divided square with two 2-simplices, one from $id_Y\circ f$ to $h$ and then one from $h$ to $g\circ id_X$.

Meanwhile, the $1$-simplices of $Hom^L(A,B)$ look like the right-top triangle in this square, with a 2-simplex from $id_Y\circ f$ to $h$, and similarly the $1$-simplices of $Hom^R(A,B)$ look like the left-bottom triangle in this square, now with a 2-simplex from $h$ to $g\circ id_X$.

**How should one think about this fact, intuitively?** Say, is there some illuminating example illustrating what goes wrong for morphism spaces compared to pinched morphism spaces?

**Second**, I'm also wondering about whether this problem might be "rectifiable" in the following sense: is there an appropriate notion of "weak equivalence" of $(\infty,2)$-categories making the following statement true?

> Given any $(\infty,2)$-category $C$, there exists an $(\infty,2)$-category $C'$ such that
>   1. The morphism spaces of $C'$ are $\infty$-categories;
>   2. The $(\infty,2)$-categories $C$ and $C'$ are weakly equivalent.

**Edit:** As noted in Tim Campion's excellent answer, the appropriate notion of morphism spaces for $(\infty,2)$-categories should be the right adjoint to the Gray tensor product. However, the morphism spaces defined by Lurie don't seem to be *too* far from the correct Hom, as they already encompass lax squares to some extent.