What is the image of the intial object inside the final object called?

Question

A recent project has forced me to consider a rather special object in a rather nasty category. Consider any category $\mathcal{C}$ which has

• image objects, meaning for each morphism $f: x \to y$ there exists an object $\text{im }f$ along with morphisms $$x \stackrel{s}{\twoheadrightarrow}\text{im }f \stackrel{i}{\hookrightarrow} y$$ so that $s$ is surjective, $i$ is injective and $f = i\circ s$, plus the obvious universal property,

• an initial object $\iota$ so that each $x \in \mathcal{C}_0$ admits a unique morphism $\iota \to x$, and

• a final object $\phi$ so that each $x \in \mathcal{C}_0$ admits a unique morphism $x \to \phi$.

Note that in my case $\iota \neq \phi$, so there is no zero object in $\mathcal{C}$. What we do have instead, is a unique morphism $\iota \to \phi$. This morphism has an associated image object, which I will label $d$.

Has $d$ been defined and studied somewhere? What are its fundamental properties?

I'm sorry if the question is seen as an obvious reference request. Category theory seems particularly rich in this Rumplestiltskin phenomenon, where the difference between finding what you are looking for and flailing around miserably lies simply in knowing a special name.

Answer 1

One context where an object like your $d$ has been studied at least a bit is in quasitopos theory. I doubt this will be very helpful to you since your "nasty" category is probably not a quasitopos, but perhaps it is interesting and may give some ideas.

Unlike in a topos, in a quasitopos $E$ the map from the initial object $0$ to the terminal object $1$ need not be a strong monomorphism (the relevant notion of "subobject" for a quasitopos). If we factor this map as an epimorphism $0\to d$ followed by a strong mono $d\to 1$, then the slice category $E/d$ and the co-slice category $d/E$ are both quasitoposes (these are two of the four subcategories identified by Kevin in the comments), and additionally $E/d$ is a preorder (hence a Heyting prealgebra) while $d/E$ has disjoint coproducts. Moreover, $E$ is the Artin gluing of these two quasitoposes along the functor $F(A) = A^d + d$ from $E/d$ to $d/E$; see A2.6.7 of Sketches of an Elephant.