# Functoriality of base change

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

$g^*\circ f_!$.

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.

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Yes. Inject $f_ {!}$ into $f_ {\ast}$ to convert it into a claim concerning equality among "base change morphisms" (not generally isomorphisms) relating topological pushforward and pullback. Then it is a matter of just looking at the definitions (and how pullback is built as a sheafification of a certain presheaf).
I don't quite like to think of the pull-back functor as certain construction. I prefer to think of it as the adjoint functor of push-forward $f_*$ . Can you kindly give some reference about proof? Thank you very much. –  Jiuzu Hong Jun 22 '10 at 17:48
It is really straitforward by using the explicit construction of pull-back(Still I don't want to call it definition). We can construct the base change by adjunction morphisms $id\rightarrow f_*\circ f^*$ and $f^*\circ f_*\rightarrow id$, but it's not easy to check in this point of view. In general, of course I can formulate some general abstract nonsense statement, maybe it's too much and useless. –  Jiuzu Hong Jun 23 '10 at 10:51