@John: I was thinking about your \$$\mathrm{Cat}/X\$$ example in #14 – I can see how it works, and how it generalises both equivalence relations on \$$X\$$ (ie groupoid preorders with object set \$$X\$$) and monoids (ie categories with object set \$$\mathbf{1}\$$).

However, it struck me that \$$\mathrm{Cat}/X\$$ is a kind of awkward category whose definition necessarily involves talking about identity of objects (ie a necessary evil, if you like!). It looks a bit like the comma category \$$(\mathrm{Obj}\downarrow X)\$$, where \$$\mathrm{Obj}\$$ and \$$X\$$ are functors from \$$\mathrm{Cat}\$$ to \$$\mathrm{Set}\$$, but that resemblance is misleading - the coproduct in that category is straightforward - we just put the two \$$X\$$-labelled categories side-by-side.

This makes me think that the "alternating, inductive" construction arises from merging objects. The crucial point is that we have a red arrow, and a blue arrow, then we merge the red target with the blue source. This creates a new composable pair, red-then-blue.

We see this in isolated form in the standard counterexample of a functor whose "image" is not a category. Take \$$C\$$ with 4 objects and arrows \$$C_0 \rightarrow C_1\$$ and \$$C_2 \rightarrow C_3\$$, and \$$D\$$ with 3 objects and arrows \$$D_0 \rightarrow D_1 \rightarrow D_2\$$. We have a functor \$$F : C \rightarrow D\$$ such that \$$F(C_0) = D_0\$$, \$$F(C_1) = F(C_2) = D_1\$$, \$$F(C_3) = D_2\$$. Then \$$F(C_0 \rightarrow C_1)\$$ and \$$F(C_2 \rightarrow C_3)\$$ are composable in \$$D\$$, but \$$D_0 \rightarrow D_2\$$ is not in the image of \$$F\$$. So the full image of \$$F\$$ is strictly larger than the image.