Canonicity of the push-pull map for sheaves of sets This question is Vakil 2.7.F (in the December 2022 version of the notes).
Suppose we have the following commutative diagram of topological spaces:
$$
  \require{AMScd}
  \begin{CD}
    W              @>{\beta'}>>  X \\
    @V{\alpha'}VV                @VV{\alpha}V \\
    Y              @>>{\beta}>   Z
  \end{CD}
$$
Suppose also we have a sheaf $\mathscr{F}$ on $X$. Then, we define two maps from $\operatorname{hom}(\mathscr{F}, \mathscr{F})$ to $\operatorname{hom}(\beta^{-1}\alpha_*\mathscr{F}, \alpha'_*\beta'^{-1}\mathscr{F})$. Given $f: \mathscr{F} \to \mathscr{F}$, we get $\beta'^{-1}f: \beta'^{-1}\mathscr{F} \to \beta'^{-1}\mathscr{F}$. We may then take the transpose of this map, apply $\alpha_*$, apply commutativity, and finally take the transpose again to get the desired map (see Vakil for more details). Dually, you may also apply $\alpha_*$ first to get a second map. Call these maps $h_1, h_2$.
Vakil 2.7.F asks us to show that $h_1(\operatorname{id})=h_2(\operatorname{id})$. You could obviously do this by passing to components and keeping track of all the functor applications and transposes, but this is super messy. Is there a nice, abstract nonsense proof of this fact?
 A: Here is an abstract nonsense proof, based on the observation that we are constructing a natural transformation of functors from $\beta^{-1}\alpha_*$ to $\alpha’_*\beta’^{-1}$, usually called the base change homomorphism. The component of this natural transformation for $\mathscr{F}$ is exactly the image of the identity under either of the $h_i$.
Build the natural transformation as the composition of: $$\beta^{-1}\alpha_* \rightarrow \beta^{-1}\alpha_*\beta’_*\beta’^{-1}\rightarrow \beta^{-1}\beta_*\alpha’_*\beta’^{-1}\rightarrow \alpha’_*\beta’^{-1}.$$
Here the first map is the unit, middle is the commutativity isomorphism, and last is counit map. Then to see that the component on $\mathscr{F}$ is either of the maps you described applied to $\text{Id}$, one needs to sit and stare at this diagram for a while.
I’ll do one half of this, viewing things as natural transformations of functors, imagine we are building this transformation up via pieces. Start with the identity natural transformation of the identity, apply $\beta’^{-1}$ then take mates. This gives the unit for $\beta’$, then apply $\alpha_*$, and apply the commutativity iso. Then apply $\beta^{-1}$, and take mates again to compose with the final counit. This shows your first map applied to the identity is the component of this natural transformation. For the other one, interpret the other map, and get to the same composition, showing that these are equal.
