TITLE: Pushouts of equivalences of categories
QUESTION [7 upvotes]: If $f:C\to D$ is an equivalence of categories that is injective on objects, then every pushout of $f$ is also an equivalence.  This follows, for instance, because such a functor is an acyclic cofibration in the canonical model structure on Cat.
Suppose conversely that every pushout of $f$ is an equivalence of categories (in the terminology suggested by Karol here, $f$ is an "acyclic flat" functor).  Does it follow that $f$ is injective on objects?  (I'd be equally happy with an answer to the corresponding question in Gpd.)

REPLY [6 votes]: Every flat functor is injective on objects.
In this blog post Chris Schommer-Pries proves that there is a unique model structure on $\mathsf{Cat}$ with categorical equivalences as weak equivalences. The argument of his "Somewhat Less Trivial Lemma" also shows that if there is a flat functor that is not injective on objects, then $E \to *$ is also flat where $E$ is the walking isomorphism. All we need to know is that flat functors are closed under composition, pushout and retracts.
Now, let $C_2$ be the group of order $2$ and let $E \to C_2$ classify the nontrivial element. The resulting pushout of $E \to *$ is $C_2 \to *$ which is not an equivalence and hence $E \to *$ is not flat.