A quick question.



> 
> Let $j : {\sf cat} \to {\sf Cat}$ be the inclusion of small categories in locally small categories; is $j$ a dense functor?
> 
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> 


If it is not (as I expect, since it's hard for a generic large category $X$ to be determined by the canonical colimit over $(j\downarrow X))$), is it true that there is a natural isomorphism
$$
\int^{A\in \sf cat}{\sf Cat}(A,PA')\times A\cong PA' 
$$
(of course, the coend/Kan extension on the left might not exist due to size issues; but I guess it's clear what I have in mind)?


If it's not clear what I have in mind, here's the full story: provided $\text{Lan}\_jj$ exists, it is the density comonad of $j$: it is a comonad, thus it has a counit $\sigma : \text{Lan}\_jj\Rightarrow 1$ which is invertible iff $j$ is dense. 



> 
> Even though $\sigma$ is not invertible, is the whiskering $\sigma \* P$ an isomorphism (where $P$ sends a small category to its presheaf category)?
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> 
> 



Is the embedding ${\sf cat}\subset {\sf Cat}$ dense?
Yes it is. In fact, the full subcategory of $Cat$ on the walking commutative triangle is dense. So $cat$, being a full supercategory of a dense category, is dense.


