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](https://mathoverflow.net/questions/175885/morphisms-every-pushout-of-which-is-a-weak-equivalence), $f$ 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**.)



Pushouts of equivalences of categories
Every flat functor is injective on objects.


In this [blog post](http://sbseminar.wordpress.com/2012/11/16/the-canonical-model-structure-on-cat/) 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.


