For a category $\mathcal{C}$, let $\mathcal{C}-Set$ denote the category of functors $\mathcal{C}\to{\bf Set}$. Recall that given a functor $F\colon\mathcal{B}\to\mathcal{C}$, the ``composition with $F$" functor is denoted $F^*\colon\mathcal{C}-Set\to\mathcal{B}-Set.$ It has a left and a right adjoint, $F_!$ and $F_*$. I call these functors the pullback, the left pushforward, and the right pushforward.

Let $p\colon A\to B$ be a function of sets, thought of as a functor $P\colon[1]\to{\bf Set}$, where $[1]$ is the "free-arrow category," $[1]="\bullet\to\bullet$." Suppose one wants to find the image of $p$, but he or she can only use pull-backs, left pushforwards, and right pushforwards to manufacture it. In other words, suppose one wants to find a zigzag of functors $[1]=:C_0\leftarrow C_1\rightarrow C_2\leftarrow C_3\rightarrow\cdots\rightarrow C_n=[0]$ such that if we perform a pullback along all leftward functors and either a left pushforward or a right pushforward along rightward functors, then the end result will be the image set $im(p)$ of $p$ (considered as a functor $[0]\to{\bf Set}$).

This can be done. To do it, I used a sequence of the form $$[1]\leftarrow C_1\rightarrow C_2\rightarrow [0].$$ If the functors are denoted (left to right) by $F,G,$ and $H$, I found that $H_! \circ G_*\circ F^\ast (P)=im(P)$.

I'm not going to bore you with the details of $C_1, C_2$ and $F,G,H$.

Here's the question. I've seen things like $H_! \circ G_*\circ F^\ast$ before in the context of polynomial functors. Unfortunately, I don't know enough about them to know if there's a connection. Is there?

I also don't know if I can get the whole epi-mono factorization somehow. I haven't worked that long at it, but suppose I want not to end up with the set $im(p)$ but instead the maps $A\to im(f)\to B$. Can I achieve that by use of pullbacks and pushforwards as above (with $C_n=[2]$ now)? Is there any rhyme or reason to such constructions?

Thanks.