A group can be regarded as a category with one object in which all arrows are isomorphisms. As a set, the group corresponds to the set of morphisms, and the group operation corresponds to the composition of morphisms. But I see two ways of defining multiplication - one is $g\cdot h=g\circ h$ and the other is $g\cdot h=h\circ g$. Is there a "correct one"? My source (https://arxiv.org/pdf/1612.09375.pdf 1.1.8(c)) ignores this issue and just says that $\cdot $ corresponds to $\circ$. I suppose a given group corresponds in a non-unique way (in two different ways) to a category described above, am I right?
Remark: The present question arose from the following exercise: Proving $G\simeq G^{op}$ (I suppose one has to make a choice in that exercise.)
Yes, you are right, you can choose either of these two ways to interpret composition in the associated category. However, note that the two resulting categories are naturally isomorphic, where the isomorphism is given by the inverse operation in the group.