$\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}$There are (at least) seven kinds of morphism spaces for a simplicial set $X$:

- The left-pinched morphism space $\Hom^L_X(x,y)$,
- The right-pinched morphism space $\Hom^R_X(x,y)$,
- The (non-pinched) morphism space $\Hom_X(x,y)$,
- The simplicial set $\Hom_{\mathfrak{C}[X]}(x,y)$ where $\mathfrak{C}[X]$ is the rigidification of $X$,
- The simplicial set $\Hom_{\mathfrak{C}^{nec}[X]}(x,y)$ where $\mathfrak{C}^{nec}[X]$ is the Dugger-Spivak rigidification of $X$,
- The simplicial set $\Hom_{\mathfrak{C}^{hoc}[X]}(x,y)$ where $\mathfrak{C}^{hoc}[X]$ is the simplicial set defined in page 17 of Dugger-Spivak's Rigidification of quasi-categories,
- The simplicial set $\Hom^E_X(x,y)$ defined in page 15 of Dugger-Spivak's Mapping spaces in Quasi-categories.

Do these mapping spaces all agree on the "underived level"? I.e. do we have
$$
\pi_0\Hom^L_X(x,y)\cong\pi_0\Hom^R_X(x,y)\cong\pi_0\Hom_X(x,y)
$$
$$\cong\pi_0\Hom_{\mathfrak{C}[X]}(x,y)\cong\pi_0\Hom_{\mathfrak{C}^{nec}[X]}(x,y)\cong\pi_0\Hom_{\mathfrak{C}^{hoc}[X]}(x,y)
$$
$$
\cong\pi_0\Hom^E_X(x,y)
$$
for an *arbitrary* simplicial set $X$?

In particular, is this true in the special case where $x=y$? (~~Such a restriction helps already for the left/right pinched morphism spaces: by Tag 01KZ, we have $\Hom^L_X(x,x)\cong\Hom^R_X(x,x)^\mathrm{op}$, and thus $\pi_0\Hom^L_X(x,x)\cong\pi_0\Hom^R_X(x,x)$ since $\pi_0(X^\mathrm{op})\cong\pi_0(X)$.~~ **Edit: this is wrong, see R. van Dobben de Bruyn's answer below**)

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