Let $\mathcal{C}$, $\mathcal{D}$ be two small categories. Let $f\: : \: \mathcal{C}\to \mathcal{D}$ be a functor. Then it induces a functor
$$
f^{\*}\: : \: sPsh(\mathcal{D})\to sPsh(\mathcal{C})
$$
via $f^{\*}(X)=X\circ f$. Now consider $sPsh(\mathcal{D}), sPsh(\mathcal{C})$ equipped with the projective model structure. Then $f^{\*}$ is a right Quillen functor and its left adjoint is given by the left kan extension 
$$
f\_{!}\: : \: sPsh(\mathcal{C}) \to sPsh(\mathcal{D}).
$$
Assume that $X$ is a cofibrant replacement of the final functor $\*\in sPsh(\mathcal{C})$. Then 
$f\_{!}(X)$ is again cofibrant.


Q: When is $f\_{!}(X)$ again a cofibrant replacement of $\*$?



Does the kan extension preserves contractible presheaves?
$f\_!(X)$ is a cofibrant replacement of $\*$ if and only if the comma category $f/d$ has a weakly contractible classifying space, for all $d\in D$.


Indeed, the value of $f\_!(X)$ on $d$ is weakly equivalent to the homotopy colimit of the constant diagram with value $\*$ on the comma category $f/d$, because the restriction functor $sPsh(C)\to sPsh(f/d)$ is left Quillen. Such a homotopy colimit is weakly equivalent to the nerve of the indexing category.


