Consider the category where objects are spherical fusion categories and morphisms are spherical functors (preserving cups and caps). I am wondering whether there is some kind of image factorisation possible in this category, similar to finite groups, where every homomorphism can be written as a composition of an epimorphism and a monomorphism.

Consider the category where objects are strict spherical fusion categories and morphisms are strict spherical functors (preserving cups and caps). I am wondering whether there is some kind of image factorisation possible in this category, similar to finite groups, where every homomorphism can be written as a composition of an epimorphism and a monomorphism.