In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states:

Indeed, the UMP of pullbacks essentially states that composition along any function α is left adjoint to pullback along α.

However, what I could only prove, is that the adjunction holds in slices categories, whereas the pullback is still defined in the raw category. I wanted to find the adjunction with categories coherent with that of the pullback, but I failed.

Here is what I mean.

First, let name objects and arrows according to the figure below:

Let $\mathrm E$ be the category where objects belong. In this category, the pullback of $\alpha$ along $h$ is $\alpha'$ and $h'$ with the universal property of pullbacks.

To investigate how pullback's left adjoint can be $\alpha$ composition, we should search for the natural isomorphism:

$$ \begin{array}{r c l} \alpha \circ f & \rightarrow & h \\\\ \hline f & \rightarrow & pullback(\alpha, h) \\ \end{array} $$

where we saw that $pullback_{\mathrm E}(\alpha,h)$ is the couple of arrows of $\mathrm E^1$ $(\alpha',h')$. But to be consistent with the fact that $f$ is a sole arrow and not a couple, we shoud see $pullback_{\mathrm E}(\alpha,h)$ as the result of applying $\alpha$-pullback operation to $h$, so this is $pullback_{\mathrm E}(\alpha)(h)=h'$.

But, then, $\alpha \circ f$, $h$, $f$ and $h'$ are all arrows: really, if we search for arrows between those arrows, we are not in ${\mathrm E}$ anymore, we are in some kind of arrow category. However, both the couples $(\alpha \circ f, h)$ and $(f, pullback_{\mathrm E}(\alpha)(h))$ are couples where the two arrows have the same codomain. And it suffices to take slice-category definitions of arrow. Then we have $g : \alpha \circ f \rightarrow h$ and $u : f \rightarrow h'$, that will let the natural isomorphism fulfill the definition of the universal property for pullbacks.

So, finally, the natural isomorphism would be meant:

$Hom_{Slice_{C}}(\alpha\circ f, h) \cong_{(f,h)} Hom_{Slice_{A}}(f,pullback_{\mathbf E}(\alpha)(h))$

Somebody could help me see why the pullback and the adjunction do not live in the same categories? Is that what Awodey meant, or is there something wrong in my understanding?