sheaf theory - Functoriality of base change

Let $a:Wrightarrow X$, $c:Xrightarrow Z$, $b:Wrightarrow Y$ and $d:Yrightarrow Z$ 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, $d:Yrightarrow Z$, $f:Zrightarrow U$, $e:Yrightarrow V$ and $g:Vrightarrow U$. Then we have another natural isomorphism $kappa'$ between $e_! circ d^*$ and

$g^*circ f_!$.

By the universal property of pull-back, one can see that $a:Wrightarrow X$,$f circ c:Xrightarrow U$, $ecirc b:Wrightarrow V$ and $g:Vrightarrow U$ 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.