Let $N(-/\mathcal C)\colon \mathcal C\to \mathbf{sSet}$ be the functor sending $c\in\mathcal C$ to the nerve of the coslice category $c/\mathcal C$.


Given a functor $K\colon\mathcal{C}\to \mathcal{D}$, I'm trying to prove that
$$
\text{Lan}\_KN(-/\mathcal C)\cong N(-/K)
$$
where RHS is a functor $\mathcal D\to \mathbf{sSet}$ sending $d\in \mathcal D$ to the nerve of the category $d/K$ having objects $\{d\to Kc\mid c\in\cal C\}$ and obvious morphisms.


No clue about a solution: coend juggling for $\text{Lan}\_K$ gives no simplifications; the universal property of $\text{Lan}\_K$ seems likewise elusive.



$ \text{Lan}_KN(-/\mathcal C)\cong N(-/K) $
I have a vague memory that there is a slick way to see this but I've forgotten. (If I had to guess, perhaps the idea is to swap the roles played by the weight $d/K(-)$ and the diagram $-/\mathcal{C}$ and then apply the coYoneda lemma. You probably have to pass to bisimplicial sets to make this work.) In any case, a direct computation - using the standard pointwise formula for left Kan extensions - isn't too difficult.


Colimits in $\mathbf{sSet}$ are taken pointwise, so it suffices to compute the left Kan extension of $N(-/\mathcal{C})\_n \colon \mathcal{C} \to \mathbf{Set}$ for each $n$. By definition, the value of this left Kan extension at $d$ in $\mathcal{D}$ is:


$$\int^{c \in \mathcal{C}} \coprod\_{\mathcal{D}(d,Kc)} \coprod\_{c\_0\cdots c\_n} \mathcal{C}(c,c\_0) \times \cdots \times \mathcal{C}(c\_{n-1},c\_n).$$
where the interior coproduct is over all $n+1$ tuples of objects of $\mathcal{C}$. This set is isomorphic to:
$$ \coprod\_{c\_0\ldots c\_n} \int^{c \in \mathcal{C}} \mathcal{D}(d,Kc) \times \mathcal{C}(c,c\_0) \times \cdots \times \mathcal{C}(c\_{n-1},c\_n)$$ By the coYoneda lemma, this coend is isomorphic to
$$\coprod\_{c\_0\ldots c\_n} \mathcal{D}(d,Kc\_0) \times \mathcal{C}(c\_0,c\_1) \times \cdots \times \mathcal{C}(c\_{n-1},c\_n),$$
which is the set of $n$-simplices in the nerve of the slice category $d/K$.


