What is the name for this weak version of a 2-categorical pullback?
The solid square commutes up to a not necessarily invertible 2-morphism $\eta$ and has the following universal property: For any square $(P', \tilde f, \tilde g, \tilde \eta)$ as indicated in the diagram, where $\tilde \eta : g \circ \tilde f \Rightarrow f \circ \tilde g$, the 1-category of "comparison data" has a terminal object.
The objects of this category are tuples $(h,\alpha,\beta)$ consisting of a morphism $h : P' \to P$, a 2-morphism $\alpha : \tilde f \Rightarrow f' \circ h$, and a 2-morphism $\beta : g' \circ h \Rightarrow \tilde g$ such that $\alpha$ is an isomorphism (but $\beta$ might not) and such that $\tilde\eta = \beta \circ (\eta h) \circ \alpha$.
Motivation: The topos of sheaves on $\mathrm{Spec}(A)$ is the pullback of $\mathrm{Set}$ and $\mathrm{Set}[\mathrm{LocRing}]$ over $\mathrm{Set}[\mathrm{Ring}]$ in the 2-category of toposes in this sense, as detailed over at MO.
