Duality in category theory

Remarks

Two notions are dual when the defining statement of one is the dual of the defining statement of the other. Example: "$f$ is a monomorphism" says that $f \circ g_1 = f \circ g_2$ implies $g_1 = g_2$. Its dual says that $g_1 \circ f = g_2 \circ f$ implies $g_1 = g_2$, which is "$f$ is an epimorphism". Hence the fact "a composite of monomorphisms is a monomorphism" gives at once "a composite of epimorphisms is an epimorphism". Likewise a section and a retraction are dual, as in section and retraction. Non-example: the dual of a statement true in one particular category need not be true in that category. In the category of sets, every morphism $X \to \emptyset$ is an isomorphism, since $X$ must then be empty. The dual statement says that every morphism $\set{0} \to X$ out of the one-point set is an isomorphism, and it fails for $X = \set{0, 1}$. Duality in this sense is different from the duality of vector spaces, although the dual space functor turns linear maps around.