With given category $\mathcal{C}$ and its objects $A$ and $B$, a hom-set $\hom_\mathcal{C}(A, B)$ is the collection of all morphisms from $A$ to $B$. There is also a related notion of hom-functor from $\mathcal{C}$ to $\mathcal{Set}$, which map objects into related hom-sets.
But why the name "hom"? Who was the first to use this terminology and in what context?