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 $D$?

<a href="https://math.stackexchange.com/questions/4287897/proving-that-the-singular-simplicial-set-is-a-kan-complex">Through a previous thread</a> 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?