Suppose I have a morphism $f\colon X\to Y$ in some category. I want to ask for something like "the object that contains the minimum amount of information about $X$ needed to reproduce $Y$."
My guess is that the right way to do that is to ask for an object $A$ and morphisms $f_0\colon X\to A$, $f_1\colon A\to Y$, such that $X\xrightarrow{f_0}A\xrightarrow{f_1}Y = X\xrightarrow{f}Y$ and such that $A$ is universal, in that for any other object $B$ equipped with similar arrows there is a unique $h$ making this diagram commute:
Does this have a name? (Or is there another better way to talk about "the minimal factorisation of $f$"?)
This is a self-answer by the OP, summarising comments by @vanxoo, @IsAdisplayName and me.
My definition as written doesn't work, because we just end up with $A=X$, $f_0=f$ and $f_1 = \mathrm{id}_Y$.
However, what I probably want is the coimage, which is the same as what I wrote, except that we impose the condition that $f_0$ has to be an epimorphism (as does the morphism $X\to B$).
In a sufficiently set-like category, if we have an epimorphism $X\twoheadrightarrow B$ we can think of it as a coarse-grained version of $X$. A factoring $X\twoheadrightarrow B\to Y = X\xrightarrow{f}Y$ gives us a coarse-graining of $X$ that still contains "enough information" to reconstruct $Y$ (as a function of $X$ according to $f$). The coimage is the terminal object in the category of such coarse-grainings, so it's essentially the coarsest coarse-graining that still contains the information needed to reconstruct $Y$.
Alternatively, we can ask for the dual concept, image, which has a similar definition except that we require $f_1$ to be a monomorphism instead, and we ask for the initial object instead of the terminal one. In Set and similar categories this gives us the image of $f$ as a function. The two concepts often coincide.