Conjugate vertices in a graph^{1} or conjugate elements of a group^{2} are equivalent (indistinguishable, essentially the same) in *one* specific structural sense.

Isomorphic objects in a category are equivalent in *another* specific structural sense.

In both cases we don't look *inside* the objects but declare them equivalent from the *outside*.

In both cases equivalence has to do with isomorphism: with structure preserving maps between the structure the conjugate elements live in and itself (automorphisms) resp. with iso arrows between the elements themselves.

Define two objects $A,B$ in a category to be conjugate when there is an isomorphism endofunctor $F$ with $F(A) = B$.

Question 1: Is it true that any two isomorphic objects are conjugate (since there is an isomorphism endofunctor that permutes them)?

The reverse is most certainly false: There are categories with conjugate objects that are **not** isomorphic. E.g. the graphs

in the category of graphs over two fixed vertices (with graph homomorphisms as morphisms) are two such objects (#9 and #6 in the diagram below). Note that there is no morphism at all between these two graphs.

Question 2: Might it be the case that whenever two objects are conjugate-but-not-isomorphic there is no morphism between them? Or is this true only in special categories and/or special cases?

Question 3: How "normal" is it that a category contains conjugate-but-not-isomorphic objects?

Most of all I'd like to know how to think about this bewildering pair of equivalences in general terms.

## Appendix

Here is the complete category of graphs over two fixed vertices and an arrow whenever there is a graph homomorphism. Compositions and identities are omitted.

The numbers are derived from the adjacency matrices: 0 = 00|00, 1 = 10|00, ..., 15 = 11|11.

## Footnotes

^{1} $x,y$ are conjugate iff there is a $g \in \text{Aut}(G)$ with $g(x) = y $.

^{2} $x,y$ are conjugate iff there is a $g \in G$ with $gx= yg$.