I have a question about the statement of Lemma 5.5.2.1 in Lurie's [`Higher Topos Theory'](http://www.math.harvard.edu/~lurie/papers/HTT.pdf). 


``Let $S$ be a small simplicial set, let $f: S\rightarrow \mathcal{S}$ be an object of $\mathcal{P}(S^{op})$, and let $F: \mathcal{P}(S^{op})\rightarrow \widehat{\mathcal{S}}$ be the functor corepresented by $f$. Then the composition 
$$S\overset{j}{\rightarrow}\mathcal{P}(S^{op})\overset{F}{\rightarrow}\widehat{\mathcal{S}}$$
is equivalent to $f$.
"


The Yoneda embedding $j$ should be from $S$ to $\mathcal{P}(S)$, but the last line doesn't seem to be just a typo. Do I miss something? Thanks!



A question about HTT Lemma 5.5.2.1
The middle term should be $\mathcal{P}(S^{op})^{op}$. The Yoneda embedding gives a functor $S^{op} \to \mathcal{P}(S^{op})$. The functor corepresented by $f \in \mathcal{P}(S^{op})$ is given by $Hom\_{\mathcal{P}(S^{op})}(-,f)$, which is a functor $\mathcal{P}(S^{op})^{op} \to \widehat{\mathcal{S}}$.


As a reality check, the analogous theorem in ordinary category is that the composite
$$
x \mapsto Hom\_S(x,-) \mapsto Nat(Hom\_S(x,-),f)
$$
is naturally isomorphic via the Yoneda lemma to the functor $x \mapsto f(x)$.


