  The inclusion of 1-categories into $(\infty,1)$-categories fails to preserve
colimits in general, and pushouts in particular. In this note, we observe that
if one functor in a span of categories belongs to a certain
previously-identified class of functors, then the 1-categorical pushout is
preserved under this inclusion. Dwyer maps, a kind of neighborhood deformation
retract of categories, were used by Thomason in the construction of his model
structure on 1-categories. Thomason previously observed that the nerves of such
pushouts have the correct weak homotopy type. We refine this result and show
that the weak homotopical equivalence is a weak categorical equivalence. We
also identify a more general class of functors along which 1-categorical
pushouts are $(\infty,1)$-categorical.
