Functors $F\colon C\to D$ preserve commutative diagrams: if a diagram $A$ commutes in $C$, then $FA$ commutes in $D$ (where $FA$ is the diagram in $D$ that can be obtained from $A$ by applying to every object and morphism in that diagram the functor $F$). However, in general, the converse is not true: functors don't necessarily reflect the commutativity of diagrams.

What is true is that equivalences of categories both preserve and reflect commutative diagrams diagrams, i.e., one can without problems translate back and forth between the categories $C$ and $D$.

I wonder: Given a pair of adjoint functors (say $F\dashv U$, where $F\colon C\to D$ and $U\colon D\to C$) between $C$ and $D$, how much can we translate back and forth between $C$ and $D$?

Through a previous thread I came to the following conclusion: a diagram \begin{array}{cc} \,\,\,\,\,\,\,A \\ \\ i \downarrow & \,\,\,\,\,\,\,\searrow \,{f} \\ \\ A' & \xrightarrow{r} & UB \end{array} in $C$ commutes if and only if the diagram \begin{array}{cc} \,\,\,\,\,\,\,FA \\ \\ Fi \downarrow & \,\,\,\,\,\,\,\searrow \,{\alpha(f)} \\ \\ FA' & \xrightarrow{\alpha(r)} & B \end{array} commutes in $D$, where $\alpha$ denotes the natural bijection $\hom(-, UB)\cong \hom(F-,B)$.

I admit this is not exactly "reflection of diagrams" in the above sense, since only on one arrow we apply $F$ while on the other two we apply $\alpha$.

Question: Can this observation be generalized? Which kind of diagrams are preserved and reflected by adjunctions and in which sense?