+-- {: .num_defn} ###### Definition Let $F:C\to D$ be a $V$-enriched functor of $V$enriched categories, let $j:C\to [C^{op},V]$ be the $V$-enriched Yoneda embedding. The [[Yoneda extension|V-enriched Yoneda extension]] of $F$ - i.e. the [[Kan extension|left Kan extension]] $\Lan_j F$ of $F$ along $j$ is also called **realization functor** associated to $F$ and in this context denoted by $||_F$. The functor $N:\begin{cases}D^{op}\to V\\c\to j(c)\circ F^{op}\end{cases}$ is called **[[nerve]] functor** associated to $F$. =-- +-- {: .num_example} ###### Example Let $\Delta_C:\Delta\to C$ be a cosimplicial object of $C$. $$ N(A)_n :=C(\Delta_C[n],A) $$ =-- +-- {: .num_prop} ###### Proposition $$(||_F\dashv N_F)$$ =--