Is the functor $\mathsf{Cat} \to \mathsf{Cat}$, $\mathcal{C} \mapsto \mathcal{C}^{\mathrm{op}}$ representable? By this I mean: Is there a small category $\mathcal{P}$ together with isomorphisms of categories $\alpha_{\mathcal{C}} : \mathrm{Hom}(\mathcal{P},\mathcal{C}) \to \mathcal{C}^{\mathrm{op}}$ which are natural in $\mathcal{C} \in \mathsf{Cat}$?
Some of my thoughts.
What is a little bit strange here is that $\mathsf{Cat} \to \mathsf{Cat}$, $\mathcal{C} \mapsto \mathcal{C}^{\mathrm{op}}$ is not a $2$-functor: If $f,g : \mathcal{C} \to \mathcal{D}$ are two functors and $\alpha : f \to g$ is a morphism of functors, we get a morphism of functors $\alpha^{\mathrm{op}} : g^{\mathrm{op}} \to f^{\mathrm{op}}$ in the other direction. Thus, we only get a $2$-functor when we reverse the $2$-morphisms in, say, the target $\mathsf{Cat}$. But this is not the case for $\mathrm{Hom}(\mathcal{P},-)$, so that we cannot formulate any reasonable compatibility of $\alpha$ with $2$-morphisms.
But the question still makes sense if we view $\mathsf{Cat}$ as a $1$-category which is enriched over itsself, I think.
Somehow I cannot believe that $\mathcal{P}$ exists, but I couldn't disprove it so far. Basically, it is an easy task to reconstruct a category $\mathcal{P}$ from the functor $\mathrm{Hom}(-,\mathcal{P})$, for example $\mathrm{Hom}(1,\mathcal{P})\cong\mathrm{Ob}(\mathcal{P})$ and $\mathrm{Hom}(\{0<1\},\mathcal{P})\cong\mathrm{Mor}(\mathcal{P})$. But I don't know how to reconstruct $\mathcal{P}$ from $\mathrm{Hom}(\mathcal{P},-)$.
I have checked that $\mathcal{C} \mapsto \mathcal{C}^{\mathrm{op}}$ preserves coproducts and I also believe that it preserves coequalizers (can someone confirm this?).
Assume the functor is representable in some sense, and let $C$ be a small category which has an initial object but not a final one. Then $Hom(P,C)$ will have an initial object, and $C^{opp}$ won't.