Construction
Create a free object, then mod out by some relations to get some smaller, more useful object. Essentially, you take your original object, blow it up into a much bigger object via free construction, then quotient it down to an object that’s bigger than your original one, but smaller than the free object.
Examples
I’m blanking out right now, but I’ve run into more than just these 2.
Tensor Product of \(V\) and \(W\)
Quotient the free vector space generated by \(V \times W\) so bilinearity holds.
Wedge product
Same as above, but with anti-symmetry enforced as well.