I am not an expert, thus I apologize if my question is very naive. Let $M$ be a model category (I do not assume any functoriality on the factorization) 

> **Q1.** Is there a reference where it is proven that (all) homotopy colimits exist?

> **Q2.** Are homotopy colimits characterized by any universal property as in the non-homotopical case (possibly involving the Homotopy category)?

Let me invite the reader to not underestimate completely this question, under various assumptions on the model category this theorem is proved in many sources, yet I did not manage to find the most general version (if any exists).

I invite anyone to contribute with an answer about the state of art about this question. There is an incredible amount of sources, coming from different decades, thus it is not completely trivial to gather a global and coherent picture.