Possible adjoint to Yoneda embedding and Repeated Yoneda embedding? While thinking about Yoneda embedding, I came up with following two questions (I should apologies, if those are too vague):


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*Does the Yoneda embedding $y :\mathcal{C}\to\mathbf{Set}^{\mathcal{C}^{\mathbf{op}}}$ has a left adjoint? If there is a such adjoint, how does it work?
Since $y$ preserves limits, according to adjoin functor theorem, it can be a right adjoint. But so far, I haven't see anything like this. 

*The Yoneda embedding is never an equivalence between categories (so a proper embedding). Now suppose we iterate this process $$y_0=\mathcal{C}\to y_1=\mathbf{Set}^{\mathcal{C}^{\mathbf{op}}}\to y_2=\mathbf{Set}^{y_1^{\mathbf{op}}}\to y_3=\mathbf{Set}^{y_2^{\mathbf{op}}}\to\cdots.$$ This is a diagram in 2-category $\mathbf{CAT}$ indexed by the ordinal $\omega.$ Does this diagram has (a sort of) colimit? If so, how does it looks like?

 A: Regarding 1, a few points, some just gathering remarks from the comments: such categories-when locally small, at least-are called total. Unless they’re posets, they can’t be small; they must be cocomplete but in fact more is true, since not every presheaf on a large category is a small colimit of representables and cocompleteness refers only to small colimits. Being total is intuitively the requirement that $C$ should admit every large colimit that it plausibly could. Essentially all naturally occurring cocomplete large categories are in fact total, although the dual notion of cototality is less common; such categories satisfy the perfect form of the co-adjoint functor theorem, namely that any cocontinuous functor out admits a right adjoint. One explanation for categories like Grothendieck toposes and Grothendieck abelian categories which satisfy both versions of the perfect adjoint functor theorem is that they are lucky enough to be cototal, which is intuitively a weak form of having a cogenerator.
