category of Anima as presentable category

Note that \(\mathrm{Fun}(\Delta^0, X) \cong X\) in the 1 category of infinity categories.
Furthermore \(\mathrm{Fun}(\Delta^0, X)\) is an infinity groupoid by inner hom of a kan complex as kan complex.

Hence it follows as

\begin{align*} \mathrm{map}_{\mathrm{An}}(\Delta^0, X) \cong& \mathrm{core}(\mathrm{Fun}(\Delta^0,X)) \\ \cong& \mathrm{Fun}(\Delta^0,X) \\ \cong& X \end{align*}

(cf. hom kan complex of a kan enriched category equivalent to the mapping space of the homotopy coherent nerve)

which shows that

\begin{align*} \mathrm{map}_{\mathrm{An}}(\Delta^0, -) \cong \mathrm{id}_{\mathrm{An}} \end{align*}