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.