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.