The \(\infty\)-category \(\underline{\nabla_2 \otimes \mathbb E_0}\) underlying the unital \(\infty\)-operad \(\nabla_2 \otimes \mathbb E_0\) is equivalent to the ‘walking span’ 
, i.e. the pushout \([1] \sqcup_{[0]} [1]\).
Proof.
By adjunction, for any \(\infty\)-category \(\mathcal C\), we have \[\mathrm{Hom}_{\mathrm{Cat}_{\infty}}(\underline{\nabla_2 \otimes \mathbb E_0}, \mathcal C) \simeq \mathrm{Hom}_{\mathrm{Op}^{\mathrm{un}}}( \nabla_2 \otimes \mathbb E_0, \mathcal C_{\sqcup}) \simeq \mathrm{Hom}_{\mathrm{Op}} (\nabla_2, \mathcal C_{\sqcup}).\] The latter space explicitly unpacks to the space of triples \(X, Y, Z\in \mathcal C\) with maps \(X\rightarrow Y \leftarrow Z\) and hence 
. ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2