Franklin's Notes

Opposite category

The opposite category $\mathsf{C}^\text{op}$ of a category $\mathsf{C}$ is defined as the category whose objects are the same objects of $\mathsf{C}$, and which has a morphism $f^\text{op}:Y\to X$ for each morphism $f:X\to Y$ of $\mathsf{C}$. For each object $X$, the morphism $1_X^\text{op}$ serves as its identity morphism in $\mathsf{C}^\text{op}$. Composition is defined by letting $f^\text{op}g^\text{op}=(gf)^\text{op}$. Intuitively, the opposite category can be thought of as the category in which the directions of all of the morphisms are "reversed", so that the domain of each morphism becomes the codomain of its opposite morphism, and vice versa.


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