I'm wondering if there is an established terminology for the following variation on the notion of image. Given a functor $F : C \to D$, one can define the following category:
- the objects are those of $C$,
- the morphisms from $c$ to $c'$ are the morphisms from $f: F(c) \to F(c')$ in $D$ such that there exists $g:c\to c'$ in $C$ satisfying $F(g)=f$,
- the identities and compositions are inherited from $D$.
Is there a name for this category? (If $F$ is full, we recover the full image of $F$.) This category, let us call it $\mathfrak{im}(F)$ in this post, comes with a factorization of $F$ as $$ C \overset p \to \mathfrak {im}(F) \overset i \to D $$ where $p$ is full (and bijective on objects) and $i$ is faithful. However, I'm not sure this factorization is universal in any sense.