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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.

Thank you

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What do you have in mind? «The opposite category of the category of graphs» sounds like a great description of the opposite category of the category of graphs to me :-) – Mariano Suárez-Alvarez Oct 15 '12 at 9:01
What kind of description are you looking for? You just gave a very clear description of it. – Ittay Weiss Oct 15 '12 at 9:01
Since the category of graphs is a concrete one (in the technical sense), its opposite ccategory is concrete (because there is a faithful functor $\mathsf{Set}^{\mathrm{op}}\to\mathsf{Set}$, the powerset functor) This gives a realization of sorts... – Mariano Suárez-Alvarez Oct 15 '12 at 9:20
If I'm not mistaken, the category of graphs is a locally finitely presentable category but not a preorder, so its opposite cannot be locally finitely presentable. This means there is no hope of finding a description of it as a category of algebraic structures of some kind. – Zhen Lin Oct 15 '12 at 10:04
You should make precise what you mean by «such a thing», as otherwise it is impossible to know exactly what you want. – Mariano Suárez-Alvarez Oct 15 '12 at 19:02

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. $zRz' \iff z'=f(z)$.

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