In the book Joy of Cats, the authors write (page 118),
One of the nice insights that can be gleaned from category theory is that the formation of quotient structures (such as groups of cosets and identification topologies) can be viewed as the dual of the formation of substructures.
I can sort of vaguely realize this notion of duality although I am not sure whether this is a right way to view this.
In my opinion, one can view the formation of substructures as "dividing the structure into different parts" whereas one may view the process of "gluing together different parts of the structure". However, as I remarked at the beginning of this paragraph, this looks very much vague to me and I was wondering whether there is any way by which we can make the duality (as stated in the quote) formal. So my question is,
Is there any way to make this notion of "duality" more precise and formal?
In general, "$P$ is dual to $Q$" means is that an instance of $Q$ in a category $C$ is the same as an instance of P in $C^{op}$, the opposite category. So, in the case you're asking about, once you make your notion of substructure and quotient structure more precise (see here, for example), you can check that process of taking subobjects in some category $C$ is equivalent to taking quotient objects in $C^{op}$.