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.