Let me quote Barry Mazur from his essay When is one thing equal to some other thing?:
“mathematical objects [are] determined by the network of relationships they enjoy with all the other objects of their species”
I always took this slogan literally, but recently I was overcome by doubts whether I missed something.
Have a look at the toy 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.
Among others, the graphs #6 and #9 are not isomorphic (i.e. "essentially the same") but indistinguishable with respect to the network of relationships they enjoy (i.e. conjugate):
graph #6 graph #9
What is the grain of salt I have to take Mazur's slogan with? Or is there something wrong with my toy category?