TITLE: What is difference between working with small and large category of spaces?
QUESTION [7 upvotes]: The following construction have always bugged me. This is p328, Remark 5.1.6.1 in Lurie's Higher Topos Theory. Lurie begins with the following:

Construction: Let $C$ be a simplicial set.  $S$ denote the $\infty$-category of small spaces.  $s$ a vertex. Composing the Yoneda embedding $$j:C \rightarrow P(C)$$ by the evaluation map $$e_s: P(C)=Fun(C^{op}, S) \rightarrow  Fun(\{ s \}, S)\simeq S$$
We obtain the map $j_s:=e_s\circ j:C \rightarrow S$. Where $j_s$ is referred to as the functor correpresented by $s$.

The remark states that we should replace $S$ by $\widehat{S}$ the large $\infty$-category of spaces.

I agree this is what we do 1-categorically when $S$ is replaced with sets.
Q1. But provided this construction makes sense, what's wrong with not replacing it?
Q2. Now suppose we are to work with it $\widehat{S}$. Is it safe to regard $Fun(C,S)$ as a fullsubcategory $Fun(C,\widehat{S})$? What do we know about the inclusion $S \hookrightarrow \widehat{S}$  i.e. if this inclusion is

limit/colimit preserving
conservative

?

(Unfortuantely I am unable to work through the material after this - which may answer the above questions)

REPLY [6 votes]: The Yoneda embedding $y: S \rightarrow P(S)$ is informally given by $y(s)(s^{\prime}) = map_{S}(s^{\prime}, s)$ (some care must be taken if $S$ is not an $\infty$-category in interpreting the mapping space, but Lurie gives a precise definition). If $S$ is a small simplicial set, then $map_{S}(s, s^{\prime})$ is a small space, so the functor does in fact land in presheaves of small spaces, as claimed. This is not necessarily true if $S$ is a large simplicial set, and this is what I believe Lurie's remark is about. What is perhaps slightly confusing is that you only really need $S$ to be locally small (ie. to have small mapping spaces), rather than to be small itself, and that's actually a rather common occurrence.

The inclusion $S \hookrightarrow \widehat{S}$ preserves all small limits and colimits. The same will be true for $Fun(C, S) \hookrightarrow Fun(C, \widehat{S})$, because (co)limits in functor $\infty$-categories are pointwise.