In general, when it's said "factoring through" it means that some morphish took an implicit roundabout which depends on exact object and unique morphism from/into it.
As an example, let's take a look at ho operator where both categories and functor are Arrow:
In this case you can use this operator as left-to-right Arrow composition:
Sometimes we want to ignore Arrow output and replace it with some value instead, we have an operator for it - hu :
This hu operator actually works similarly to ho - also maps covariantly on 2/2 argument - but factoring through Unit object.
Let's focus on source morphisms for both ho and hu operators since it's the only difference between them:
We can use the fact that value can be transformed into a dumb Arrow:
Now watch closely at my hands - we are going to map terminal morphism contravariantly on 1/2 argument to this source morphism:
And this is exactly source morphism of ho operator!