In Dwyer and Kan's 1980 paper on "Simplicial Localizations of Categories", they prove the following result for binary categorial sums. For a set $O$, let $O$-${\mathsf{Cat}}$ be the following category. An object of $O$-${\mathsf{Cat}}$ is a small category with $O$ as its set of objects. An morphism of $O$-${\mathsf{Cat}}$ is a functor that restricts to the identity on objects. Let $sO$-${\mathsf{Cat}}$ be the category of simplicial objects in $O$-${\mathsf{Cat}}$.

A map $A\to B$ in $sO$-${\mathsf{Cat}}$ is a *weak equivalence* if, for every two objects $X,Y\in O$, the induced map of simplicial sets ${\mathrm{Hom}}_A(X,Y)\to {\mathrm{Hom}}_B(X,Y)$ is a weak homotopy equivalence. Kan and Dwyer's result is that if two maps $A\to A'$ and $B\to B'$ in $sO$-${\mathsf{Cat}}$ are weak equivalences, then so is their categorial sum $A+B\to A'+B'$.

My question is whether this result extends to infinite categorial sums? Kan and Dwyer's approach is to show that if $A\to A'$ is a weak equivalence then $A+B\to A'+B$ is a weak equivalence. Hence their approach does not generalize to the infinite case.