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?

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)?