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Let $\mathsf{MM}^{++}(\kappa)$ state that the forcing axiom$\mathsf{MM}^{++}$ can be instantiated only for stationary set preservingposets of size at most $\kappa$. We give a detailed account of Asper\`o andSchindler's proof that $\mathsf{MM}^{++}(\kappa)+$there are class many Woodincardinals implies Woodin's axiom $(*)$ if $\Diamond_\kappa$ holds and$\kappa>\aleph_2$. Our presentation takes advantage of the notion ofconsistency property: specifically we rephrase Asper\`o and Schindler's forcingas a specific instantiation of the notion of ``consistency property'' used byMakkai, Keisler, Mansfield and others in the study of infinitary logics. Wealso reorganize the order of presentation of the various parts of the proof.Taken aside these variations, our account is quite close to the original proofof Asper\`o and Schindler.

Another proof that $\mathsf{MM}^{++}$ implies Woodin's axiom $(*)$