This means that each functor $f$ decomposes as a composition of the form $j e$, where $e$ is bijective on objects and $j$ fully faithful; and if

$\array{
A &\overset{u}{\longrightarrow}& C
\\
\mathllap{{}^{e}}\big\downarrow &&\big\downarrow \mathrlap{{}^{j}}
\\
B &\underset{v}\longrightarrow& D
}$

is a commutative diagram with $e$ bijective on objects and $j$ fully faithful, then there is a unique functor $h \colon B\to C$ such that $h e = u$ and $j h = v$. The object through which $f$ factors is called the full image of $f$.

In fact, this can be generalized to a square commuting up to invertible natural transformation, in which case one still concludes that $h e = u$ but that $j h \cong v$, with the isomorphism composing with $e$ to give the original isomorphism. This means that this is an enhanced factorization system.