Definition Suppose $X,Y,Z$ are sets, and we have maps \begin{eqnarray*} f\colon Y&\to& Z, \\ \Phi\colon X&\to& Y. \end{eqnarray*}Then the pullback of $f$ under $\Phi$ is the mapping \begin{eqnarray*} \Phi^\ast f\colon X &\to& Z, \\ x&\mapsto& (f\circ\Phi)(x). \end{eqnarray*} Let us denote by $M(X,Y)$ the set of all mappings $f\colon X\to Y$ . We then see that $\Phi^\ast$ is a mapping $M(Y,Z)\to M(X,Z)$ . In other words, $\Phi^\ast$ pulls back the set where $f$ is defined on from $Y$ to $X$ . This is illustrated in the below diagram.

Properties

1. For any set $X$ , $(\operatorname{id}_X)^\ast = \operatorname{id}_{M(X,X)}$ .
2. Suppose we have maps \begin{eqnarray*} \Phi\colon X&\to& Y, \\ \Psi\colon Y&\to& Z \end{eqnarray*}between sets $X,Y,Z$ . Then $$ (\Psi\circ \Phi)^\ast = \Phi^\ast \circ \Psi^\ast.$$
3. If $\Phi\colon X\to Y$ is a bijection, then $\Phi^\ast$ is a bijection and $$ \big(\Phi^\ast\big)^{-1} = \big(\Phi^{-1}\big)^\ast. $$
4. Suppose $X,Y$ are sets with $X\subset Y$ . Then we have the inclusion map $\iota:X\hookrightarrow Y$ , and for any $f\colon Y\to Z$ , we have $$ \iota^\ast f = f|_X, $$ where $f|_X$ is the restriction of $f$ to $X$ .