Take a proper subobject $m: A \to B$ in a locally presentable category.
Since the category is locally presentable, $B = \text{colim} B_i$ where $B_i$ are presentables. Under which hypothesis m factors through a $B_i$? Can you show me some counterexamples of when it does not?!
A simple example is to take $\text{Vect}$. $\text{Vect}$ is locally finitely presentable with generators given by the finite-dimensional vector spaces; every vector space is a filtered colimit of finite-dimensional vector spaces. But most subspaces of an infinite-dimensional vector space are also infinite-dimensional.