It is known that [fully faithful functors are closed under pushouts in Cat](https://ncatlab.org/nlab/show/full+and+faithful+functor) (e.g. Lemma 4.9 of [this paper](https://lmcs.episciences.org/742/pdf)). Are locally fully faithful 2-functors closed under (strict) 2-pushouts in the 2-category 2-Cat of 2-categories, (strict) 2-functors, and 2-natural transformations? I expect this to be true, but giving an explicit description of a 2-pushout is daunting. Is there a simpler way to reason to prove this by reasoning entirely locally (i.e. in the hom-categories)?


I expect the fully weak setting to be more difficult, but if it is known that locally fully faithful pseudofunctors are closed under pseudopushouts in a bicategory of bicategories, this would also answer my question.



Are locally fully faithful 2-functors closed under 2-pushout in 2-Cat?
No. I'm not sure whether you're asking about pushout along an arbitrary 2-functor, or just about pushing out a locally fully faithful functor along *another* locally fully faithful functor. Either way the answer is no.


Consider the inclusion of 1-categories into 2-categories as the locally discrete ones. This functor is fully faithful. It has a right adjoint, so it preserves pushouts. And it identifies the faithful functors between 1-categories with the locally fully faithful functors between essentially discrete 2-categories. So it suffices to show that faithful functors of 1-categories are not closed under pushout in $Cat$.


So it suffices to show that injective homomorphisms of monoids are not closed under pushout in $Mon$.


For this, if you're asking about pushout along an arbitrary map, then it suffices to consider the pushout of the injection $\mathbb N \to \mathbb Z$ along the map $\mathbb N \to \mathbb N / (2=1)$, which is the non-injection $\mathbb N / (2=1) \to \ast$.


Otherwise, we're pushing out an injective homomorphism along another injective homomorphism. The result need not be an injective homomorphism -- see [here](https://mathoverflow.net/questions/249589/pushouts-of-injective-monoid-homomorphisms?) (the first-linked paper at Benjamin Steinberg's answer there gives an example with $\leq 4$-element semigroups; adding disjoint unit elements these are $\leq 5$-element monoids).


