Model $\mathrm{\infty }$-categories III: the fundamental theorem

We prove that a model structure on a relative $\mathrm{\infty }$-category $(M,W)$ gives an efficient and computable way of accessing the hom-spaces $ho{m}_{M[[W^{-1}]]}(x,y)$ in the localization. More precisely, we show that when the source $x\in M$ is cofibrant and the target $y\in M$ is fibrant, then this hom-space is a “quotient” of the hom-space $ho{m}_M(x,y)$ by either of a left homotopy relation or a right homotopy relation.