Understanding model independently the equivalence of two ways of obtaining homotopy types from categories It is well known that any homotopy type can be obtained as the classifying space of a ($1$-)category. The classifying space of a category $\mathcal{C}$ can be interpreted in at least two ways: 


*

*We can view $\mathcal{C}$ as an object in the $(\infty,1)$-category of $(\infty,1)$-categories, and localise $\mathcal{C}$ along all its morphisms. 

*We can consider the constant functor $\mathcal{C} \to \mathbf{Type}, \; X \mapsto *$ (where $\mathbf{Type}$ denotes the $(\infty,1)$-category of homotopy types), and take its colimit. (Or thus equivalently the free homotopy colimit of the unique functor $\mathcal{C} \to 1$, where $1$ denotes the category with only one morphism; see Dugger).


Using various models these two views as well as their equivalence can be made precise as follows: View $\mathcal{C}$ as an object in $\mathbf{SSet}$ equipped with the Joyal model structure. Inverting all morphisms of $\mathcal{C}$ just corresponds to taking its fibrant replacement in the Kan model structure. Taking the colimit in 2. can be formalised by taking the geometric realisation of (the nerve of) $\mathcal{C}$: This is exactly the formula you get for computing the homotopy colimit of the constant functor $\mathcal{C} \to \mathbf{Top}, \; X \to *$ using the simplicial replacement of this functor (described e.g. here). 
My question then is: 

Is there a model independent way of seeing that these two constructions are equivalent? 

So I'm looking for an argument which could be formalised in any model of $(\infty,1)$-categories. 
 A: Here is an argument, which is basically Denis Nardin's comment.   
To have a model independent proof you need model independent definitions of the hocolim and of the localization. You can define them via adjunctions, but from a pragmatic point of view I am not sure this is so helpful. Ultimately to do any kind of calculation you will have to pick a model and a construction of the localization/hocolim and if your definition is the model independent one, then you have the additional task of proving that they satisfy these definitions. 
Anyway... 
There is an ($\infty$-)adjuction of $(\infty,1)$-categories: 
$$hocolim_C : Cat_{(\infty,0)}^C \leftrightarrows Cat_{(\infty,0)}: const$$
and this is a definition $hocolim_C$. Here $Cat_{(\infty,0)} \simeq Top$ is the usual $(\infty,1)$-category of spaces which the OP called Type. 
There is another one:
$$ ||-||: Cat_{(\infty,1)} \leftrightarrows Cat_{(\infty,0)}: i  $$
which defines the localization at all morphisms $||C||$. Here $i$ is the inclusion of $(\infty,0)$-cats (= spaces) into all $(\infty,1)$-categories. 
Then this means that for all spaces $X$ we have
$$ 
\begin{align}
Cat_{(\infty,0)}( hocolim_C const(*) , X) &\simeq Cat_{(\infty,0)}^C( const(*), const(X)) \\
&\simeq Fun(C, Cat_{(\infty,0)}(*, X)) \\
&\simeq Fun( C, iX) \\
& \simeq Cat_{(\infty,0)}( ||C||, X)
\end{align}
  $$
And so by Yoneda you conclude that $hocolim_C const(*) \simeq ||C||$. 
