TITLE: Why pullback only defined up-to-isomorphism but nevertheless presented as functor?
QUESTION [19 upvotes]: I am reading "Category Theory" (2nd ed.) of Awodey, and I'm stuck at page 96 (proposition 5.12) when pullbacks are presented as functors:
The pullback under question corresponds to this square:
$$\begin{matrix}
C' \times_C A & \xrightarrow{h'} & A \\[1ex] 
\downarrow \rlap{\scriptstyle{\alpha'}} & &  \downarrow\rlap{\scriptstyle{\alpha}} \\[1ex] 
C' & \xrightarrow{h} &  C
\end{matrix}$$
Here is the statement of Awodey's book that I do not understand:

Pullback is a functor. That is, for fixed $C' \rightarrow_h C$ in a category $\mathbf{C}$ with pullbacks, there is a functor
$h^* : \mathbf{C}/C \rightarrow \mathbf{C}/C'$
defined by
$(A\rightarrow_\alpha C) \mapsto (C'\times_C A \rightarrow_{\alpha'} C')$
where $\alpha'$ is the pullback of $\alpha$ along h

The problem that I see is that, given initially:
$$\begin{matrix}
& & A \\[1ex]
& & \downarrow \rlap{\scriptstyle{\alpha}} \\[1ex]
C' & \xrightarrow{h} & C
\end{matrix}$$
there can be several pullbacks on it, for example, in addition to $(\alpha',h')$, there could be $(\alpha_2',h_2')$, and the unique condition is that there exists an isomorphism $i$ such that $\alpha_2' = \alpha\circ i$ and $h_2' = h'\circ i$.
Worse, given a pullback $(\alpha',h')$, one can build as many as pullbacks as there exist isomorphisms, as given any isomorphism $j$ (with domain $C' \times_C A \rightarrow_{h'}$) the two arrows $(\alpha'\circ j,h' \circ j)$ form a new pullback.
So, how could we build a functor if the image arrow is only defined up to an arbitrary isomorphism?

REPLY [13 votes]: Why has no one mentioned anafunctors? They were invented by Makkai for the express purpose of expressing universal constructions as functor-like things (I was going to say 'objects') without having to make choices. 
The definition is as follows: Let $C$ and $D$ be categories. An anafunctor from $C$ to $D$ is a span $C\leftarrow U \to D$ where $U \to C$ is fully faithful and surjective on objects.
That's it.
Of course, it is a bit more tricky to show that there are the arrows of a (weak) 2-category of categories, and that category theory works perfectly well using this notion of categories; for the present purposes, the references at the above link should satisfy the most curious.
In the case of the pullback 'functor' $h^*$ we have an anafunctor
$\mathbf{C}/C \leftarrow P \to \mathbf{C}/C'$ where $P$ is the category with objects pullback squares with bottom arrow $h$ and morphisms the canonical thing which makes $P \to \mathbf{C}/C$ fully faithful: commutative triangles involving the other leg of the cospan involving $h$ - in other words, arrows in $\mathbf{C}/C$. This induces a canonical arrow in $\mathbf{C}/C'$ by the universal property of pullbacks, and then the functor $P\to \mathbf{C}/C'$ just forgets the original pullback square, and keeps the arrow with codomain $C'$. This is a functor by using the universal property of pullbacks a couple of times.
If one has a way of choosing a particular pullback square for each object of $\mathbf{C}/C$, say by some sort of canonical construction as in the category of (ZF-)sets (Kuratowski pairs and subsets), or by judicious amounts of the axiom of choice, then one can find a section on objects of $P \to \mathbf{C}/C$, and this gives you a canonical section of the functor $P \to \mathbf{C}/C$, which then gives a pullback functor $h^*\colon \mathbf{C}/C \to \mathbf{C}/C'$.
A similar sort of anafunctor exists for any universal construction, which is defined up to isomorphism by some universal property.