Let $B$ be a subset of the codomain of $f : X \to Y$ that contains $f(X)$. Formulate the universal property satisfied by the inclusion map $i : B \to Y$ that characterizes it up to isomorphism among all injective maps into $Y$ whose image contains $f(X)$.
I have been wondering how the universal property and the "isomorphism" should be defined here. One attempt is to define the universal property satisfied by $i$ as the following: there exists a unique map $g: X \to B$ such that $f = i \circ g$. However, this allows a lot of maps $i':B' \to Y$ to be considered isomorphic to $i$, even if the domain of $i'$, $B'$, is not the same size as $B$. Moreover, $i$ and $i'$ can have different images in this sense, which makes it harder to define an isomorphism.
I'd appreciate it if someone could guide me on defining a universal property.