Today I was explaining to my friend about opposite categories, and how the definition of monomorphism leads to defintion of epimorphism in opposite category. He asked me quite an interesting question. If we have a morphism in a category, how does that relate to the morphism in the opposite category?
For example, let's consider that category Top, and a homomorphism between two topological spaces. What would be exact expression for arrow in the opposite category?
I think it'd be something like the inverse function but I am not sure.
The answer is a bit underwhelming for some. I'm gonna give a practical answer and not get deep into philosophical aspects of this, but there are other interpretations to be given. You should consider this a simplification, but I believe it's a good one to start thinking about these concepts.
It may be convenient to think of the opposite category as the original category but where you draw the arrows backwards, and that's it.
It's just a formal device that helps us visualize and interpret certain concepts, and thus helps us with proofs as well.
We use this notion because it's easier to think this way but it shouldn't be necessary in principle. Human brains are not that powerful to do proofs without them, that's all.
In our case, if you have an arrow $f:a\rightarrow b$ in $Top$ and go to the opposite category, then $f^{op}:b\rightarrow a$ is not to be thought as a continuous function in $Top$ from $b$ to $a$, but as one from $a$ to $b$ too. It should be thought of as $f$ since nothing really changed.
Otherwise you can't really interpret it as a function. It's purely formal.