# Lifting of a morphism in a hom-functor when hom-sets are empty

I'm having trouble in unserstanding a hom-functor.

Suppose we have a hom-functor $$Hom(X,$$_$$)$$ for some Category $$\mathcal{C}$$.

Suppose further that the hom-sets Hom$$_{Set}(X,A)$$ and Hom$$_{Set}(X,B)$$ are empty and there exists an arrow $$f$$ going from $$A$$ to $$B$$.

What would be the lifted function $$\mathcal{C}(X,f)$$ in category $$Set$$?

Isnt't this a function going from the empty set to the empty set?

How does this work?

First, note that the notation for the hom-set is $$\mathrm{Hom}_{\mathcal C}(X, A)$$, rather than $$\mathrm{Hom}_{\mathbf{Set}}(X, A)$$ (the $$\mathbf{Set}$$ is implicit).
Yes, $$\mathrm{Hom}_{\mathcal C}(X, f)$$ will be a function $$\emptyset \to \emptyset$$ if $$\mathrm{Hom}_{\mathcal C}(X, A)$$ and $$\mathrm{Hom}_{\mathcal C}(X, B)$$ are empty. But that's okay! There is exactly one function from the empty set to the empty set, which is the identity function. (In fact, for each set $$Y$$, there's exactly one function from the empty set to $$Y$$. We just have to provide an assignment of each element in the domain to an element of the codomain: when the domain is empty, this is trivial, as there aren't any elements to provide assingments for.)