In this video, at just after the 5 minute mark, the speaker says:
"...the colimit of the diagram has the homotopy type of the homotopy colimit. Why? Because all of the maps included in the diagram are closed cofibrations, and there is a theorem that if you have a diagram where everything is a closed cofibration, the homotopy type of the colimit and the homotopy colimit are the same."
What is the theorem being referenced here, or alternatively how does this follow?
EDIT:
If I have the following diagram:
$$* \rightrightarrows *$$ I think that this is a diagram of closed cofibrations for which the statement does not hold, for example.
I think the speaker's statement does hold in the particular type of cases that he is talking about, and that the maps in the diagrams he speaks of are closed cofibrations, but that there are extra conditions on the diagrams which he is not mentioning. I would like any further insight that anyone can offer (even in general, about diagrams whose colimits and hocolimits are homotopy equivalent).