Suppose $\mathcal{E}$ is a subcategory of $\mathbf{Cat}$, $F, F' : \mathcal{E} \rightarrow \mathcal{E}$ and $\eta : F \Rightarrow F'$, such that each component $\eta_X$ is a weak category equivalence (that is, full, faithful and essentially surjective). How do $\operatorname{Alg}(F)$ and $\operatorname{Alg}(F')$ relate? (Besides there being a functor $\operatorname{Alg}(F') \rightarrow \operatorname{Alg}(F)$, by precomposing the algebra structure map by $\eta_X$ - however, I think that this functor is not an equivalence).
I'm pretty sure that, if $\eta_X$ are strong category equivalences, there is a strong equivalence between $\operatorname{Alg}_w(F)$ and $\operatorname{Alg}_w(F')$ (where by $\operatorname{Alg}_w(F)$, I mean category of algebras and algebra morphisms, where the usual algebra morphism square commutativity condition is weakened by a non-identity $2$-cell).
PS: Note that I do not assume AC. (So weak $\not\Rightarrow$ strong equivalence)