Let $\mathbf{C}$ be a category and $F,G:\mathbf{C}\to\mathbf{Cat}$ be category-valued functors on $\mathbf{C}$. Suppose there is a family of equivalences of categories
$$(\Phi_C:FC\simeq GC)_{C\in\mathbf{C}}\tag{1}$$
such that for all $f:C\to C'$ in $\mathbf{C}$, there is a natural isomorphism
$$\Phi_{C'}\circ Ff\cong Gf\circ\Phi_C\tag{2}$$
that is, the following diagram commutes up to natural isomorphism:
$\require{AMScd}$
\begin{CD}
FC @>{\Phi_C}>> GC\\
@V Ff V V\cong @VV Gf V\\
FC' @>>{\Phi_{C'}}> GC'
\end{CD}
Is there a standard name for such a $\Phi$ (or for something similar to such a $\Phi$)? I've looked around but haven't been able to find it.
Note it is just a generalization of a natural isomorphism for category-valued functors, which allows equivalence instead of isomorphism in (1) and natural isomorphism instead of equality in (2). It captures the intuitive notion of an equivalence of categories which is "natural" in that it respects functors between the categories.
As an example, consider the functor
$$\mathbf{Sets}^{(-)}:\mathbf{Sets}^{\mathrm{op}}\to\mathbf{Cat}\tag{3}$$
which maps a set $I$ to the functor category $\mathbf{Sets}^I$ of $I$-indexed families of sets and maps a function $f:J\to I$ to the "reindexing functor" $\mathbf{Sets}^f:\mathbf{Sets}^I\to\mathbf{Sets}^J$, and the functor
$$(-)^*:\mathbf{Sets}^{\mathrm{op}}\to\mathbf{Cat}\tag{4}$$
which maps the set $I$ to the slice category $\mathbf{Sets}/I$ and maps the function $f:J\to I$ to the pullback functor $f^*:\mathbf{Sets}/J\to\mathbf{Sets}/I$. The functors (3) and (4) are related by the above notion, which shows that the equivalence
$$\mathbf{Sets}^I\simeq\mathbf{Sets}/I$$
is "natural" in the set $I$.
Any pointers are appreciated.
Best Answer
If your commutativity natural isomorphisms are coherent with the composition and identities, then nlab calls this a pseudonatural equivalence, which you can find towards the bottom of the linked page.
Since this would otherwise be essentially a link only answer, let me add a couple comments. First of all, the natural setting for this is 2-category theory and 2-functors, so we should regard $\mathbf{C}$ as a 2-category which only has identity 2-morphisms, and then our functors become (strict) 2-functors, though if you wanted, you could now generalize to lax/oplax 2-functors.
Next, I'd like to add a point on coherence, and why we might expect it/want it. Suppose we have $f:c\to c'$, $g:c'\to c$, then we get $$ \require{AMScd} \begin{CD} Fc @>Ff>> Fc' @>Fg>> Fc'' \\ @V\Phi_c VV \cong_{f} @V\Phi_{c'}VV \cong_g @VV\Phi_{c''}V \\ Gc @>Gf>> Gc' @>Gg>> Gc'' \\ \end{CD} $$ We would expect that when we paste $\cong_f$ and $\cong_g$ together like this that we get back $\cong_{gf}$, the natural isomorphism making the outer square commute: $$ \require{AMScd} \begin{CD} Fc @>F(gf)>> Fc'' \\ @V\Phi_c VV \cong_{gf} @VV\Phi_{c''}V \\ Gc @>G(gf)>> Gc''. \\ \end{CD} $$
Otherwise, if the commutativity natural isomorphisms are arbitrary, we can't make much use of the concept, since we can't relate them with the category structure.