This question originally came up when thinking about topoi: what conditions must be imposed on a topos for the initial object to be a subobject of every other object? I will weaken the question to the following:
In an arbitrary category $C$ with initial object $0$, what further conditions must $C$ satisfy for $0$ to be a subobject of every other object?
I've worked out that if $C$ is well pointed, then the implication holds. I'll include the proof of this claim. But if you are not interested in reading the proof, please skip to the last paragraph. For any $x \in C$, let $!_X : 0 \to X$ denote the unique morphism. Let $f,g : Y \to 0$ such that $!\circ f = ! \circ g$. I am essentially asking under what conditions does it follow that $f = g$.
There are two related, though, to my understanding, distinct definitions of well pointed. The first is just that if $f\circ x = g \circ x$ for all $x : 1 \to X$, then $f = g$. The second includes the condition that $1$ is not initial (i.e. $1\not=0$). Lets work under the second definition for now. If we have $x : 1 \to 0$, then it follows that $x$ is an isomorphism (since this implication is true in any category). Thus $0$ has no points. This implies that if $f : Y \to 0$, then $Y\cong 0$. Because of the morphism $f : Y \to 0$, $Y$ cannot have any points. Thus it is vacuously true that for all points $y$ of $Y$, we have $!_Y \circ f \circ y = Id_y\circ y$. By well pointedness, we have $!_Y \circ f = Id_y$. Clearly $f\circ !_Y = Id_0$.
So, if $C$ is well pointed (under the second definition), then $!\circ f = ! \circ g$ indeed implies that $f = g$, namely because it follows that $Y$ is initial and maps out of initial objects are unique. However, I do not think this proof carries over to the first definition, since we can have maps into the initial object without the domain also being initial.
So, I all I can show is that an initial object in a well pointed category (second definition) is a subobject of every other object. But this is a fairly strong condition. In the context of topos theory, this implies all sort of Boolean properties. Yet I see no way to generalize my proof to weaker assumptions. Does anyone know of any milder assumptions that suffice? Are there any conditions that do not require $C$ to have a terminal object? I would happily accept an answer to either the original inspiration of this post (i.e. conditions on topoi), or the weaker question (conditions on arbitrary categories).