I want to understand the notion of an opposite category in a very specific case, say the category $Grp$ of groups with homomorphisms as morphisms.
In the opposite category of groups $Grp^\circ,$ for some $G$ and $G'$ in this category, the set Hom$_{Grp^\circ}(G, G')$ of morphisms between them is defined to be the set Hom$_{Grp}(G', G)$ of morphisms between $G'$ and $G$ in the category $Grp$ of groups. It is known that there exists a contravariant functor $$op: Grp \to Grp^\circ $$ defined to be the identity map. This means that the morphism $f \in Hom_{Grp}(G, G')$ will get mapped to itself. Hence, my question:
Given a group homomorphism $f: G \to G'$ between two groups $G$ and $G'$, how can we also think of $f$ as a map from $G'$ to $G$ since $f$ is not assumed to have an inverse?
This question was answered many times on this site, but every time it's about general categories. I would like to understand this idea in a particular case of the category of Groups and with homomorphisms as their morphisms. I would appreciate any help. Thanks.
It's a bit like a Zen koan, but the answer is that we simply do what we say we're doing: we think of $f:G\to G'$ as a "map" between $G'$ and $G,$ and that is how we do it.
The key thing to be sure of is that "map", here, does not mean any kind of function. It's thus better to say "morphism" or "arrow" to emphasize that the concept is purely abstract and formal. To summarize again, we think of a homomorphism $f:G\to G'$ as an arrow $G'\to G$ by reversing the order of the symbols on the page. That's what we do, and there is no more meaning to it than that.