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I went through few posts on MSE but It was not sufficent. So I am studying opposite category and I cannot understand what does it mean by by reversing a map? I know that the objects are the same and thecomposition will make sense once I understand morphisms.

Suppose $\mathfrak{C}$ is a category and if $A,B$ are two objects in $\mathfrak{C}$, then for any morphism $f : A \to B$, we define a morphism from $f^{op}:B \to A$. But how is the map $f^{op}$ defined from $f$?

I cannot find a way in which $f^{op}$ is well defined. Can anyone help me?

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  • $\begingroup$ It is defined abstractly, as an element of the set $\mathbf C^{\mathrm{op}}(B, A)$. It doesn't have any concrete meaning further than that. Dualisation is usually entirely abstract (though there are some instances when the opposite of a category also has a concrete interpretation). $\endgroup$
    – varkor
    Mar 23 at 13:25
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To specify a category, we need to specify the objects, the morphisms, the domain and codomain of each morphism, and the composition operation on morphisms. The objects and morphisms are just abstract sets (or classes); they do not have to have any internal structure themselves. (For example, there need not be any notion of "elements" of objects, and morphisms do not need to be functions or function-like entities.)

Given a category $\mathcal{C}$, the opposite category $\mathcal{C}^{op}$ is defined to have the same classes of objects and morphisms, but the domain and codomain of morphisms are reversed, as is the order of composition. So, by definition, given objects $X$ and $Y$ of $\mathcal{C}$, the set (or class) of morphisms $\operatorname{Hom}_{\mathcal{C}^{op}}(X, Y)$ is just $\operatorname{Hom}_{\mathcal{C}}(Y, X)$; that is, by definition, a morphism $X \to Y$ in $\mathcal{C}^{op}$ literally is the same thing as a morphism $Y \to X$ in $\mathcal{C}$.

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I'm kinda new too to the concept of category, but I think you are just confused. Think about the category made by the couple $\big(X, \leq\big)$ where $\leq$ is a preorder. An arrow $x\rightarrow y$ means: $x\leq y$. Now, if you move to the opposite, or rather the dual, you might have something like $\big(X^{op},\geq \big)$, and now you can interpret the dual arrow as $x\rightarrow y \iff x\geq y$. But notice that this is just a particular case. In general, Categories of Lattices should provide enough example of how to interpret the dual, but in general it is just an abstract costruction not really bounded to the original idea. You should accept that in category we have no elements so you can't search the meaning of an arrow by the way it acts on the element of an object, simply because this last sentence makes no sense in this setting. Rather, the information you have on an arrow merely relies on the category you find it, and on its "direction".

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