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Definition

Opposite Category

Category Theory · category.tex
Let $\mathcal{C}$ be a category (Definition~Category). The opposite category $\mathcal{C}^{op}$ is defined as follows.
  1. Objects: $$ \operatorname{Ob}(\mathcal{C}^{op}) = \operatorname{Ob}(\mathcal{C}) $$
  2. Morphisms: $$ \Hom_{\mathcal{C}^{op}}(X,Y) = \Hom_{\mathcal{C}}(Y,X) $$
  3. Composition is defined by reversing composition in $\mathcal{C}$.
Associativity and identity laws follow directly from those in Definition~Category.
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