That a construction is free doesn’t mean that you lose nothing. It means that if you’re going to do some construction anyway, you might as well use the free one, because the free one can get to any other. (Attainable utility anyone?)
Showing that your construction is free means that all you need to show as worthwhile is constructing any category from our quiver. Adjunctions are a fine reason, though I wish we could introduce adjunctions first and then show that we need categories to get them.
That a construction is free doesn’t mean that you lose nothing. It means that if you’re going to do some construction anyway, you might as well use the free one, because the free one can get to any other. (Attainable utility anyone?)
Showing that your construction is free means that all you need to show as worthwhile is constructing any category from our quiver. Adjunctions are a fine reason, though I wish we could introduce adjunctions first and then show that we need categories to get them.