Is the category where $\mathrm{Obj}$ has just two different elements $a$ and $b$ and the non identity arrows are $a \to b$ and $b \to a$ really a category? Does it have a name?
The thing is that the "usual" examples I've encountered so far have a lot of objects, like sets, groups, rings, topological spaces, etc... so I'm not sure if this construction leads to something that is not considered a category.
From this question I can see that there is indeed a category with two objects and precisely one arrow, but I'd like to be sure that everything is okay if we add one more arrow on the other direction.
It was confirmed in the comments that this is indeed a category. It also has a name: the free(-standing) isomorphism. Functors from this category into some category $\mathcal C$ are precisely isomorphisms in $\mathcal C$.