# A question about HTT Lemma 5.5.2.1

I have a question about the statement of Lemma 5.5.2.1 in Lurie's Higher Topos Theory'.

`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!

• The usual Yoneda embedding is from $S^{\operatorname{op}}$ to $\mathcal{P}(S)$ - to get $j$, apply the Yoneda embedding to $S^{\operatorname{op}}.$ – dhy Nov 30 '18 at 3:20
• Thanks for fixing HTT. That is my reason for the vote up. – Piotr Hajlasz Nov 30 '18 at 3:55
• By the way, we field short questions like this in the homotopy theory chat here. – Harry Gindi Nov 30 '18 at 4:31
• The second and third P(S^op)’s should have an extra op on the outside, yeah? That ought to do it. – Dylan Wilson Nov 30 '18 at 6:10
• @dhy No, this is actually a typo. The usual Yoneda embedding goes from $S\hookrightarrow \mathcal{P}(S)$. Presheaves are contravariant. Dylan's comment makes this correct. – Harry Gindi Nov 30 '18 at 8:16

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 $$x \mapsto f(x)$$.