I want to figure out when two categories of "descent data" are equivalent when we have equivalences on each "chart" commuting with the "restriction functors" up to an isomorphism of functors.
Fix an indexing set $I$ and assume we are given categories $A_i, B_i, A_{i,j}, B_{i,j}, A_{i,j,k}, B_{i,j,k}$. One can imagine $I$ to be indexing an open cover $U_i,\ i\in I$ of a topological set and the families $A$ and $B$ of categories to be categories of sheaves of some sort on $U_i$ or their double and triple intersection $U_{i,j},\ U_{i,j,k}$. Assume we are also given restriction functors labeled $l$ and $r$ with the indexing hopefully clear from the example $l^i_{i,j}:A_i\to A_{i,j}.$
Assume we have two categories of descent data $\mathrm{Desc}_1$ and $\mathrm{Desc}_2$. The objects of the first are collections $(M_i, f_{i,j})$, where $M_i$ are objects of categories $A_i$ and $f_{i,j}$ are morphisms $f_{i,j}: l^i_{i,j}M_i\to l^j_{i,j}M_j$ satisfying the cocycle condition
$$l^{i,k}_{i,j,k}f_{i,k}=l^{j,k}_{i,j,k}f_{j,k}\circ l^{i,j}_{i,j,k}f_{i,j}.$$ The objects of the second category are collections $(N_i, g_{i,j})$ defined in the same way using categories $B$ and functors $r$.
Assume I have equivalences $F_i:A_i\to B_i$ and similarly for "double and triple intersections" such that these functors commute with the restriction functors $l,r$ up to isomorphism. In particular, for objects $M_i\in A_i$ we have natural isomorphisms $F_{i,j}l^i_{i,j}M_i\to r^i_{i,j}F_iM_i.$
It seems in this setting I can define an equivalence from $\mathrm{Desc}_1$ to $\mathrm{Desc}_2$. A collection $(M_i, f_{i,j})$ is sent to a collection with objects $F_iM_i$ and the morphisms defined as compositions
$$r^i_{i,j}F_iM_i\xrightarrow{\sim}F_{i,j}l^{i}_{i,j}M_i\xrightarrow{F_{i,j}f_{i,j}}F_{i,j}l^{j}_{i,j}M_j\xrightarrow{\sim}r^j_{i,j}F_jM_j.$$
I think the right way to talk about such things is using the language of stacks and 2-categories, but I have very little experience with 2-categorical computations, so I would prefer to make do with what I have.
I think one only needs to check that the morphisms defined in this way satisfy the cocycle condition and managed to prove that, but something still bothers me. When starting out, I expected the proof to involve some additional condition on the isomorphism making the diagrams of functors commute, but it is never to be found! Is this perhaps a known fact? I don't think it is present in the literature in this exact form, but maybe someone could point me to the right statement in a more advanced language?