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Given a functor $U: \Delta \rightarrow \textbf C$ from the simplex category into $\textbf C$, I want to show that if $\mid -\mid$ is "geometric realization" of a simplicial set in $\textbf C$ given by the left Kan extension (assuming it exists)

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the diagram commutes. I.e $\mid \text{yoneda}(-)\mid = U(-)$. This is true for the topological case ($\textbf{C} = \text{Top})$ but I don't know about the general case.

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This is true in considerable generality. If $y:A\to B$ is fully faithful and $f:A\to C$, then $Lan_u f\circ u$ is naturally isomorphic to $f$ whenever $Lan_u$ is a pointwise left Kan extension, for instance, whenever $C$ is cocomplete. The reason is that $Lan_u f(b)$ is the colimit of the diagram indexed by the comma category $u/b$ whose value at $u(a)\to b$ is simply $f(a)$. When $u$ is fully faithful, $u/u(a)$ is equivalent to $A/a$, so has a terminal object $id_a:u(a)\to u(a)$, and the colimit is the value $f(a)$ taken at $id_a$.

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