There are two "opposite" functors: $$ op_\Delta\colon sSet\to sSet$$ and $$op_s\colon sCat\to sCat.$$ The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite of a functor $\Delta\to \Delta$ which is the identity on objects and takes a morphism $\langle k_0,\ldots,k_n\rangle\colon [n]\to [m]$ (where $k_i$ is the integer that $i$ gets mapped to by this morphism) to the morphism $\langle m-k_n,\ldots,m-k_0\rangle$. For example, the morphism $[1]\to [2]$ that takes $0$ to $0$ and $1$ to $1$ gets mapped to the morphism that takes $0$ to $1$ and $1$ to $2$.

The second functor takes a simplicial category to the opposite simplicial category, which is easier to define. It has the same objects but given $x,y\in C^{op}$, the mapping complex $C^{op}(x,y)=C(y,x)$.

There is also the simplicial nerve functor $N\colon sCat\to sSet$. I am interested in a proof of the fact that for a given fibrant simplicial category $C$, there is a weak equivalence of quasicategories $op_\Delta\circ N(C)\simeq N\circ op_s(C)$.

I'm relatively certain that this is an elementary proof, but I don't feel skilled enough with the simplicial nerve to figure out the details. Does anyone have a proof of this fact?