This may be a silly question, but are there any examples of categories that don't have duals (ie, neither for objects nor arrows)? I'm currently under the impression that it's impossible to not have a dual category (since for any $\mathcal C$, you can just reverse the arrows to get $\mathcal C^{op}$).
I was reading through a paper from John Baez & Mike Stay, and on page 5 they give a "chart of concepts" to "help the reader safely navigate the sea of jargon". I've been making a mind map out of this chart and have been finding examples for each one (eg, the category of sets as an example of Cartesian closed categories, the category of finite-dimensional vector spaces as an example of compact closed categories, etc).
In the process, I realized that the arrows I was drawing for my mind map were necessarily adding more structure (eg, "add 'closed'", "add 'Cartesian'", etc), and a friend pointed out that there's a "forgetful functor" in which each kind of category "forgets" that part of it's structure.
I think a part of my confusion is the issue of "adjunction" and how left adjoints don't guarantee right adjoints (and vice versa) which made me wonder if the idea of duality itself has such a characteristic.