Suppose I have a functor $$ X_\bullet: I \to \text{Spaces} $$ where $I$ is a small filtered category.

It seems to be a "folk theorem" that the homomorphism $$ \underset{\alpha\in I}{\text{colim }} H_\ast(X_\alpha) \to H_\ast(\underset{\alpha\in I}{\text{hocolim }} X_\alpha) $$ is an isomorphism, where $H_\ast$ denotes singular homology.

Is there a reference for this?

(Preferably standard?)