Given a category $C$, an opposite category $C^{op}$ is defined as
$ob(C^{op})=ob(C)$
For every $x,y\in ob(C)$, $\text{Hom}_{C^{op}}(x,y)=\text{Hom}_C(y,x)$
One can informally think of this as reversing "arrows" in category $C$.
However, in most abstract sense, morphisms are not arrows. So how does reversing arrows correspond to definition of morphisms in $C^{op}$?
Things in $\text{Hom}_C(y,x)$ have a "source" $y$ and "target" $x$. Now these things become $\text{Hom}_{C^{op}}(x,y)$, but my mind keeps going in this way: surely things in $\text{Hom}_{C^{op}}(x,y)$ would have source $x$ and target $y$, so how does something with swapped sources/targets can be in this set?
Edit: Now I think I see what this is all about. So we don't really care what these morphisms are, but the important fact is that they must satisfy some rules regarding compositions etc. So what we really have is a consistent system of things that follow axioms, and that what we call $\text{Hom}$ sets (in particular they don't have sources or targets or even have to be arrows). In opposite category, we impose exactly same system of "morphisms" but just in opposite direction.
The question has been answered multiple times on math.SE.
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