Opposite category

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Definition

Suppose $\mathcal{C}$ is a category. The opposite category to $\mathcal{C}$, denoted ${\mathcal{C}}^{op}$, is defined as follows:

• The objects of this category are the same as the objects of $\mathcal{C}$: $\mathrm{O}\mathrm{b}{\mathcal{C}}^{op}=\mathrm{O}\mathrm{b}\mathcal{C}$.
• For $A,B\in \mathrm{O}\mathrm{b}\mathcal{C}$, ${\mathcal{C}}^{op}(A,B)=\mathcal{C}(A,B)$.
• The identity maps remain the same.
• For $A,B,C\in \mathrm{O}\mathrm{b}\mathcal{C}$, with $f\in \mathcal{C}(A,B),g\in \mathcal{C}(B,C)$, the composite $f\circ g$ in ${\mathcal{C}}^{op}$ equals the composite $g\circ f$ in $\mathcal{C}$.