@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.

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.