Reading the notion of image in category theory, I've found on nLab the following one:
Let $\mathfrak{C}$ be a category and let $M \subseteq Mono(\mathfrak{C})$ be a subclass of the monomorphisms in $\mathfrak{C}$. Let moreover $f: c \to d$ be a morphism in $\mathfrak{C}$. The $(M)$-image of $f$ is the smallest $M$-subobject $im(f) \to d$ through which $f$ factors (if it exists).
Assume that there exists also the greatest $M$-subobject through which $f$ factors. Has it been named in category theory?
Doesn't sound like a very useful notion, so I doubt it has a name. If the class of monomorphisms $M$ contains the identity morphisms (which is almost always the case, if not always) then the greatest $M$-subobject through which $f:c\to d$ factors is the total subobject, $d$ itself.