Let $a:W\rightarrow X$, $c:X\rightarrow Z$, $b:W\rightarrow Y$ and $d:Y\rightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism $\kappa$ between two functors $b_! \circ a^*$ and
$d^* \circ c_!$. Usually this natural isomorphism is called base change.
Suppose we have another pull-back diagram, $d:Y\rightarrow Z$, $f:Z\rightarrow U$, $e:Y\rightarrow V$ and $g:V\rightarrow U$. Then we have another natural isomorphism $\kappa'$ between $e_! \circ d^*$ and
By the universal property of pull-back, one can see that $a:W\rightarrow X$,$f \circ c:X\rightarrow U$, $e\circ b:W\rightarrow V$ and $g:V\rightarrow U$ is also a pull-back diagram. Denote the corresponding natural isomorphism by $\kappa''$.
Is it true that $\kappa''=\kappa'\circ \kappa$?
Probably the equality is a little confusing, but the formulation is clear if one thinks of it.