In [HTT](http://www.math.harvard.edu/~lurie/papers/croppedtopoi.pdf).5.5.8.18 Lurie defines a projective object $P$ in a quasicategory $\bf C$ as an object such that its corepresented functor ${\rm Map}(P,-)$ "commutes with geometric realizations". I can catch the general idea that something like $\hom(P,|S|\_{X^\*}) \simeq |S|\_{\hom(P,X^\*)}$ should happen, if 


1. $X\_\*$ is a cosimplicial object in $\bf C$;
2. $|-|\_{\hom(P,X^\*)}$ is the "realization" obtained via familiar abstraction from the cosimplicial object $X^\*$: since $\hom(P,X^\*)\in [\Delta, {\bf Set}]$ it can be Kan-extended to a functor ${\bf sSet}\to \bf C$, and this functor admits a right adjoint (the "nerve").
3. $|-|\_{X^\*}$ is a realization **similarly** induced as a ($\infty$?)-functor $N(\boldsymbol \Delta)\to \bf C$.


But this "similarly" is kind of too vague, I strive for a more precise description. Unfortunately HTT seems to skip a precise definition of the paradigm "geometric realization"-"nerve".


Does it transport to the $(\infty,1)$-categorical case? Do I recover the *precise* form of that statement (i.e.: give a cosimplicial object in $\bf C\in QCat$, then there is a pair of $(\infty,1)$-adjoint functors obtained as Yoneda extension / "nerve")?


On the same vein, Remark 5.5.8.5 is totally obscure to me:



> 
> The formation of the geometric realizations of simplicial
>  objects should be thought of as the $\infty$-categorical analogue of the formation of reflexive coequalizers.
> 
> 
> 


Can you help me?



Projective objects in HTT
The term geometric realization is used in HTT to refer to colimits indexed by $\Delta^{op}$. So an object $P \in \mathcal{C}$ is projective if and only if, for every simplicial object $X\_{\ast}$ in $\mathcal{C}$, the canonical map
$$ \varinjlim \text{Map}(P, X\_{\ast} ) \rightarrow \text{Map}(P, \varinjlim X\_{\ast} )$$
is a homotopy equivalence of spaces. 


If $\mathcal{C}$ is an ordinary category and $X\_{\ast}$ is a simplicial object of $\mathcal{C}$, then a colimit of $X\_{\ast}$ is a coequalizer of the pair of face maps $X\_1 \rightarrow X\_0$. Since there is also a degeneracy map $X\_0 \rightarrow X\_1$, this is a reflexive coequalizer. Many good properties of reflexive coequalizers in ordinary category theory generalize to statements about geometric realizations in $\infty$-categories.


