# Which diagrams are reflected by adjunctions?

Functors $$F\colon C\to D$$ preserve 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 diagrams.

What is true is that equivalences of categories both preserve and reflect 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?

Instead of talking about reflecting diagrams, let's talk about reflecting equality of arrows. After all, to say a diagram commutes is to say that any two paths through it (i.e., two arrows from one object in the diagram to another) are equal. To say that a functor $$F:C\to D$$ reflects equality of arrows is to say that each map $$F\colon \mathrm{Hom}_C(A,B)\to \mathrm{Hom}_D(F(A),F(B))$$ is injective. That is, I think you're asking when adjoint functors are faithful.

One answer is that the left adjoint $$F$$ is faithful when the unit $$\eta\colon \text{Id}_C\to UF$$ is a monomorphism, and the right adjoint $$G$$ is faithful when the counit $$\varepsilon\colon FU\to \text{Id}_D$$ is an epimorphism. And these conditions are frequently satisfied!

Given arrows $$f,g\colon A\to B$$ in $$C$$, such that $$F(f) = F(g)$$, we also have $$GF(f) = GF(g)$$, and we get a naturality square: $$\require{AMScd} \begin{CD} A @>{f,g}>> B\\ @V{\eta_A}VV @VV{\eta_B}V \\ GF(A) @>{GF(f) = GF(g)}>> GF(B) \end{CD}$$ Now $$\eta_B\circ f = GF(f)\circ \eta_A = GF(g) \circ \eta_B = \eta_B\circ g$$, so if $$\eta_B$$ is monic, we get $$f = g$$.

The argument for the counits is dual, using the naturality square: $$\begin{CD} FG(A) @>{FG(f)=FG(g)}>> FG(B)\\ @V{\varepsilon_A}VV @VV{\varepsilon_B}V \\ A @>{f,g}>> B \end{CD}$$

In the general case, we still get that if $$F(f) = F(g)$$, then $$f$$ and $$g$$ are coequalized by $$\eta_B$$, and if $$G(f) = G(g)$$, then $$f$$ and $$g$$ are equalized by $$\varepsilon_A$$.

• I'm not sure faithfulness is exactly the question here. The example in the problem holds for any adjunction, right? Commented Nov 9, 2021 at 16:46
• @KevinArlin True, I guess I answered the question "which adjoints reflect diagrams?", not "which diagrams are reflected by adjoints?". But the question as stated is fairly open-ended, and I think the observations in my answer might be useful/interesting to the OP. Commented Nov 9, 2021 at 21:30