I was reading Steve Awodey's book Category Theory and one concept I really do not understand.
On the page $94,$ we have
In $\textbf{Sets},$ take a function $f: A \to B$ and a subset $V \subset B.$ Let, as usual, $$f^{-1}(V) = \{a \in A | f(a) \in V\} \subset A$$ and consider $$\require{AMScd} \begin{CD} f^{-1}(V) @>\bar{f}>> V \\ @V{j}VV @VV{i}V \\ A @>>{f}> B. \end{CD}$$ where $i$ and $j$ are canonical inclusions and $\bar{f}$ is the evident factorization of the restriction of $f$ of $f^{-1}(V)$ (since $a \in f^{-1}(V) \implies f(a) \in V$)
My confusion rises on the factorization of the restriction. I do not know the definition of it. Moreover, in the book, Awodey mentioned that diagram is related to a pullback. I am curious about why the diagram is a pullback.
If $a\in f^{-1}(V)$, then $f(a)\in V$. So $\bar f:f^{-1}(V)\to V$ is well-defined by $\bar f(a)=f(a)$.
To see the square is a pullback, consider a commutative square $$\require{AMScd} \begin{CD} X @>g>> V \\ @V hVV @VV{i}V \\ A @>>{f}> B. \end{CD}$$ If $x\in X$, then $f(h(x))=i(g(x))\in V$, so that $h(x)\in f^{-1}(V)$. So there is a map $k:X\to f^{-1}(V)$ defined by $k(x)=h(x)$. One then checks that $h=j\circ k$ and $g=\bar f\circ k$, and that $k$ is the unique map with these properties.