Word for “least factor” in category theory

Question

Given a morphism $f:A \to B$, I would like to express the fact that $f$ factors through every morphism $f':A' \to B$ (not necessarily uniquely). Is there a common expression for this (and the dual)?

Answer 1

So I understand $f$ factoring through every other such morphism $f'$ as follows: for every $f': A'\to B$ there is some $a: A\to A'$ s.t. $f = f'\cdot a$. This property of $f$ means that $f: A\to B$ is the weakly initial object of the slice category $\mathcal{C} / B$.

I'm not sure "how much" you want to dualize. If you revert all the arrows you obtain of course the weakly final object of the coslice category $B/\mathcal{C}$.

If we're dualizing it to an $f: A\to B$ through which every $f': A'\to B$ factors, then we have the weakly final object of the slice category.