In mathematics, it is very common to turn one object into another by removing some structure. For example, you may turn a field into a ring and then into an abelian group if you ignore division and multiplication. I believe *forgetting* is the standard term here.

It is also possible to *add* some structure (though mathematicians normally do not like this and only do it when there is a very good reason). For example, you may turn the euclidean plane $\mathbb{E}^2$ into the Cartesian product
$\mathbb{R}^2$ if you promote a point into the origin and draw two lines through it. Or you can make a smooth manifold into a Riemannian one.

I have two questions about this stuff. Is there a standard name for this operation? And, what is the correct (coherent, conceptual, etc) treatment of it? I have no clue, for example, how to properly describe it in categorical terms (as opposed to the former case where it is a very well known forgetful functor).

anyfunctor $F: A \to B$ can be considered to be a "forgetful functor": ncatlab.org/nlab/show/stuff,+structure,+property Echoing what Ben just said, just as $F$ can be considered a way in which to forget some of the stuff/structure/properties adhering to (objects of) $A$, it could also be viewed as a way in which to add stuff/structure/properties adhering to (objects of) $B$, parallel to how we say that a structure $X$ (in the sense of model theory) is a reduct of another structure $Y$ iff $Y$ is an expansion of $X$. $\endgroup$