According to the n-lab article on 2-pullbacks,
A 2-pullback in a 2-category is a square $$ \begin{array}{ccc} P & \xrightarrow{p} & A \\\ q\downarrow & & \downarrow{f}\\\ B & \xrightarrow{g} & C \end{array} $$ which commutes up to isomorphism, and which is universal among such squares in a 2-categorical sense. This means that
(1) given any other such square $$ \begin{array}{ccc} Z & \xrightarrow{v} & A \\\ w\downarrow & & \downarrow{f}\\\ B & \xrightarrow{g} & C \end{array} $$ which commutes up to isomorphism, there exists a morphism $u:Z→P$ and isomorphisms $pu≅v$ and $qu≅w$ which are coherent with the given ones above, and
(2) given any two morphisms $u,t:Z→P$ and 2-cells $α:pu→pt$ and $β:qu→qt$ such that $fα=gβ$ (modulo the given isomorphism $fp≅gq$), there exists a unique 2-cell $γ:u→t$ such that $pγ=α$ and $qγ=β$.
My question is, if $\alpha$ and $\beta$ are isomorphisms, does $\gamma$ also need to be an isomorphism? If not in general, are there some conditions on $p$ and $q$ (weaker than $p$ and $q$ being equivalences) that would make it true?
Specifically, I have a situation where $p$ has a left quasi-inverse, i.e. $sp\cong id$ for some $s$.