aspect a bit later in these notes, but an elementary introduction to this theory can be found in .) Finally Bill Dwyer, Dan Kan and Justin Smith, [@DKS], introduced a similar construction for an $\mathbb{A}$ which is an $\mathcal{S}$-category to start with, and motivated it by saying that $\mathcal{S}$-functors with domain this $\mathcal{S}$-category corresponded to *$\infty$-homotopy commutative $\mathbb{A}$-diagrams*, yet they do not seem to be aware of the history of the construction, and do not really justify the claim that it does what they say. Their viewpoint is however important as, basically, within the setting of Quillen model category structures, this provides a cofibrant replacement construction. Of course, any other cofibrant replacement could be substituted for it and so would still allow for a description of homotopy coherent diagrams in that context. This important viewpoint can also be traced to Grothendieck’s *Pursuing Stacks*, [@stacks].

The DKS extension of the construction, [@DKS], although simple to do, is often useful and so will be outlined next. If $\mathbb{A}$ is already a $\mathcal{S}$-category, think of it as a simplicial category, then applying the $\mathcal{S}$-construction to each $\mathbb{A}_n$ will give a bisimplicial category, i.e. a functor $S(\mathbb{A}) : \mathbf{\Delta}^{op} \times \mathbf{\Delta}^{op} \to Cat$. Of 