No functor to subcategory?

Question

As far as I’m aware, you can always craft a function from one set to another unless the target is the empty set. To me this is easy to understand because the empty set is...empty, so of course you can’t draw a map on it. But I’m a bit less clear on the same thing with categories. Say you have a category with objects A and B, and morphisms $f: A \to B$ and $g: B \to A$. There would be no functor from this category to a wide subcategory with only $f$ and not $g$ because you would a) lose composition and b) be mapping the hom-set of morphisms from B to A to the empty set. But to me this is just a bit weird that you don’t have a functor from a category to a subcategory of it since it feels like there is an obvious logical relationship from the category to the subcategory in question. It feels like there should be a functor. Am I making a mistake somewhere, or is this something I just have to get used to?