Let's try to pin down the idea of the opposite category, which is a kind "mirror image" of a catgory, obtained by systematically reversing "left" and "right" in the construction of a category. Where do left and right appear?

Consider a morphism \\(f: A \rightarrow B\\). We may view this as a relationship between \\(A\\) and \\(B\\), which object \\(A\\) on the left and object \\(B\\) on the right. In a mirror image category, we could have a mirror-twin morphism \\(f'\\), with the labelling reversed, so that the left object of \\(f'\\) is \\(B\\) and the right object is \\(A\\).

The standard names for 'left' and 'right' are 'domain' and 'codomain'.

The reversal is stated as follows:

\\[dom(f') = cod(f)\\]

\\[cod(f') = dom(f)\\]

The opposite \\(X'\\) of a category \\(X\\) has the same objects as \\(X\\), and its system of morphisms consists of all the twins of the morphisms in \\(X\\). (The standard nomenclature is \\(X^{op}\\)).

Now suppose that \\(f: A \rightarrow B\\), \\(g: B \rightarrow C\\) in \\(X\\). This is a composable pair, and we may form the composition \\(f \triangleright g\\).

To distinguish composition in the opposite category \\(X'\\), let's use a different symbol for the composition operator.

For morphisms \\(u, v\\) in \\(X'\\), we'll write \\(u \triangleleft v\\) for their composition in \\(X'\\).

Having established this notation, now let's turn our attention back to the composable pair \\(f \triangleright g\\) in \\(X\\).

The question naturally arises: how can we form a composable pair from \\(f'\\) and \\(g'\\) in the opposite category \\(X'\\)?

First let's try \\(f' \triangleleft g'\\). For that to work, as in any category, we would need that \\(cod(f') = dom(g')\\).

But \\(cod(f') = dom(f)\\), and that is not equal to \\(dom(g') = cod(f)\\). Bzzt.

The other choice for the composable pair is \\(g' \triangleleft f'\\), and this works well. Indeed, we do have that \\(cod(g') = dom(f')\\), which follows from the premise that \\(cod(f) = dom(g)\\).

In a word:

\\[g' \triangleleft f' = (f \triangleright g)'\\]

These three equations define the transformation of a category \\(X\\) into its opposite category \\(X'\\).

Consider a morphism \\(f: A \rightarrow B\\). We may view this as a relationship between \\(A\\) and \\(B\\), which object \\(A\\) on the left and object \\(B\\) on the right. In a mirror image category, we could have a mirror-twin morphism \\(f'\\), with the labelling reversed, so that the left object of \\(f'\\) is \\(B\\) and the right object is \\(A\\).

The standard names for 'left' and 'right' are 'domain' and 'codomain'.

The reversal is stated as follows:

\\[dom(f') = cod(f)\\]

\\[cod(f') = dom(f)\\]

The opposite \\(X'\\) of a category \\(X\\) has the same objects as \\(X\\), and its system of morphisms consists of all the twins of the morphisms in \\(X\\). (The standard nomenclature is \\(X^{op}\\)).

Now suppose that \\(f: A \rightarrow B\\), \\(g: B \rightarrow C\\) in \\(X\\). This is a composable pair, and we may form the composition \\(f \triangleright g\\).

To distinguish composition in the opposite category \\(X'\\), let's use a different symbol for the composition operator.

For morphisms \\(u, v\\) in \\(X'\\), we'll write \\(u \triangleleft v\\) for their composition in \\(X'\\).

Having established this notation, now let's turn our attention back to the composable pair \\(f \triangleright g\\) in \\(X\\).

The question naturally arises: how can we form a composable pair from \\(f'\\) and \\(g'\\) in the opposite category \\(X'\\)?

First let's try \\(f' \triangleleft g'\\). For that to work, as in any category, we would need that \\(cod(f') = dom(g')\\).

But \\(cod(f') = dom(f)\\), and that is not equal to \\(dom(g') = cod(f)\\). Bzzt.

The other choice for the composable pair is \\(g' \triangleleft f'\\), and this works well. Indeed, we do have that \\(cod(g') = dom(f')\\), which follows from the premise that \\(cod(f) = dom(g)\\).

In a word:

\\[g' \triangleleft f' = (f \triangleright g)'\\]

These three equations define the transformation of a category \\(X\\) into its opposite category \\(X'\\).