What I would add here is that the property of left-cancellation is exactly equivalent to injectivity, i.e., f : A -> B is injective iff, for any g, h : C -> A, f o g = f o h implies g = h. If A = {} then f is injective and left-cancellative, both vacuously.

The subtlety is now that left-cancellativity is not equivalent to having a left inverse, for exactly the reason pointed out.

The value of this observation is that left-cancellativity is a useful generalization of injectivity that works in any category, where left-cancellative morphisms are called monomorphisms. If you already know about monomorphisms, it's easier to notice that there's something "off" about D&F's exercise!