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 ${\mathcal {C}}$: $\operatorname {Ob} {\mathcal {C}}^{op}=\operatorname {Ob} {\mathcal {C}}$.
• For $A,B\in \operatorname {Ob} {\mathcal {C}}$, ${\mathcal {C}}^{op}(A,B)={\mathcal {C}}(A,B)$.
• The identity maps remain the same.
• For $A,B,C\in \operatorname {Ob} {\mathcal {C}}$, 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 ${\mathcal {C}}$.