Duality in category theory

Summary

Duality in category theory is the correspondence between a statement about a category and the statement obtained by reversing all arrows. A statement holds in the opposite category $\mathcal{C}^{\mathrm{op}}$ exactly when its dual holds in $\mathcal{C}$. So a theorem proved for all categories gives, for free, its dual theorem. Pairs of dual notions include monomorphism and epimorphism, initial and terminal objects, product and coproduct, and limit and colimit.