Does anyone know where I can find a description of the opposite category of the category of graphs? The morphisms of the category are graph homomorphisms.
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What about considering a subcategory of the category of graphs and all binary relations $R$ between them (such that if edge $e:x-y$ is in relation with $e':x'-y'$ then $xRx'$ and $yRy'$, too)? Specifically, collect those relations $R:G-G'$ which are inverses of a graph morphism $f:G'\to G$, i.e. $zRz' \iff z'=f(z)$.