Opposite category $\mathcal C^{\mathrm{op}}$

For a category $\mathcal C$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), the opposite category $\mathcal C^{\mathrm{op}}$ has the same objects and reverses every morphism:

$\mathcal C^{\mathrm{op}}(A,B)=\mathcal C(B,A).$

The identity at $A$ remains $1_A$. If $f^{\mathrm{op}}:B\to A$ and $g^{\mathrm{op}}:C\to B$ correspond to $f:A\to B$ and $g:B\to C$ in $\mathcal C$, define $f^{\mathrm{op}}\circ_{\mathrm{op}}g^{\mathrm{op}}=(g\circ f)^{\mathrm{op}}$. Associativity and the identity laws follow directly from those of $\mathcal C$, so this prescription is a category. Moreover $(\mathcal C^{\mathrm{op}})^{\mathrm{op}}=\mathcal C$ strictly.