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We relate the relative nerve $\mathrm{N}_f(\mathcal{D})$ of a diagram ofsimplicial sets $f \colon \mathcal{D} \to \mathsf{sSet}$ with the Grothendieckconstruction $\mathsf{Gr} F$ of a simplicial functor $F \colon \mathcal{D} \to\mathsf{sCat}$ in the case where $f = \mathrm{N} F$. We further show that anystrict monoidal simplicial category $\mathcal{C}$ gives rise to a functor$\mathcal{C}^\bullet \colon \Delta^\mathrm{op} \to \mathsf{sCat}$, and that therelative nerve of $\mathrm{N} \mathcal{C}^\bullet$ is the operadic nerve$\mathrm{N}^\otimes(\mathcal{C})$. Finally, we show that all the aboveconstructions commute with appropriately defined opposite functors.

The Operadic Nerve, Relative Nerve, and the Grothendieck Construction