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.)


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