ct.category theory - First: upper-star, then: lower-star, finally: lower-shriek

First of all: yes, there's certainly a connection. See http://ncatlab.org/nlab/show/polynomial+functor. If the base category is $Set$, the composite

$$Set/W stackrel{f^ast}{to} Set/X stackrel{g_ast}{to} Set/Y stackrel{h_!}{to} Set/Z$$

first takes a $W$-indexed set $S_w$ to an $X$-indexed set $T_x = S_{f(x)}$, then takes this to the $Y$-indexed set $U_y = prod_{x: g(x) = y} T_x$, then takes this to the $Z$-indexed set $V_z = sum_{y: h(y) = z} U_y$. Putting this together, the composite is a family of polynomials, each a sum of monomial terms

$$P(ldots, S_w, ldots) = (z mapsto sum_{y in h^{-1}(z)} prod_{x in g^{-1}(y)} S_{f(x)})$$

I'll give a quick example. Suppose we want to express the free monoid functor

$$F(S) = sum_{n geq 0} S^n$$

in this form. Then we take $W = 1$, $X = mathbb{N} times mathbb{N}$, $Y = mathbb{N}$, $Z = 1$. There's only one choice for $f$ and $h$, and $g$ is rigged so that the fiber over $n in mathbb{N}$ is an $n$-element set: $g(m, n) = m + n + 1$. One can easily check this works.

As for the other question: it would have been nice if you had "bored" us! Because I don't see how to reconstruct what you did. What I have to get the image is a zig-zag of length 4

$$Set^{[1]} stackrel{F^ast}{to} Set^{C_1} stackrel{G_ast}{to} Set^{C_2} stackrel{H^ast}{to} Set^{C_3} stackrel{J_!}{to} Set^{[0]}$$

where $C_1$ is the generic cospan $a to c leftarrow b$, $C_2$ is the generic commutative square, $C_3$ is the generic span $a leftarrow d to b$, and then $G$ and $H$ are the evident inclusion functors, and $F$ takes each arrow of the generic span to the arrow of $[1]$. Then $F^ast$ takes $p: A to B$ to the cospan consisting of two copies of $p$; hitting this with $G_ast$ takes this cospan to the pullback square (pulling back $p$ against itself); hitting this with $H^*$ restricts the pullback square to the span consisting of the pullback projections; finally, hitting this with $J_!$ takes this span to its colimit = pushout, which is the same as the coequalizer of the pullback projections (because they have a common right inverse). (Based on his comment, I'm guessing that some guy on the street was doing more or less the same thing.)

Could you tell us what you had in mind?