Let $F: \mathcal C \to \mathcal D$ is a functor with the following property. Any morphism $f: F(X) \to F(Y)$ in $\mathcal D$ factors only through objects mapped by $F$. In other words there is no object $Z \notin$ image of $F$, through which $f$ could be factored.
Does this functor property have some common name?
UPDATE
As an example let's consider a category $\mathcal C$ of data types (boolean, natural number, integer, real, ...) and a subtype relation defined on types. Let $F$ be a functor $Set$, which maps data types to corresponding set types (set of booleans, set of natural numbers, ...). For example there is a morphism $natural < real$ in $\mathcal C$. This morphism is mapped to the morphism $Set(natural) < Set(real)$. The last one can be factored only through $Set(integer)$. All morphisms between $Set(natural)$ and $Set(real)$ belongs to the image of $Set$.