> [!definition] > > Let $\catc$ be a [[Category|category]], $Z \in \obj{\catc}$. Let > $ > \obj{\catc_Z} = \bigcup_{X \in \obj{\catc}}\mor{X, Z} > $ > and for any $f: X \to Z$ and $g: Y \to Z$, the morphisms in $\mor{f, g}$ are the morphisms in $h \in \mor{X, Y}$ such that $f = g \circ h$. > > $\catc_Z$ is then is the category of **objects over** $Z$. > [!definition] > > The [[Product|product]] of $f: X \to Z$ and $g: Y \to Z$ in $\catc_Z$ is known as the **fibre product** $f$ and $g$ in $\catc$. > > More concretely, let $(f \times_Z g, \pi_f, \pi_g)$ be their product where $f \times_Z g: X \times_Z Y \to Z$, $\pi_f: f \times_Z g \to f$, and $\pi_g: f \times_Z g \to f$. Identifying them as morphisms in $\catc$, > $ > \begin{CD} > X \times_Z Y @>{\pi_g}>> Y \\ > @V{\pi_f}VV @VV{g}V \\ > X @>>{f}> Z > \end{CD} > $ > then $f \times_Z g = g \circ \pi_g = f \circ \pi_f$. Here, $\pi_f$ is the **pullback** of $g$ by $f$, which can also be denoted as $f^*(g)$. > > If $S \in \obj{\catc}$, and $u_X: S \to X$ and $u_Y: S \to Y$ are morphisms such that $f \circ u_X = g \circ u_Y$, then there exists a unique morphism $u: S \to X \times_Z Y$ such that > $ > f \circ u_X = f \circ \pi_f \circ u = g \circ \pi_g \circ u = g \circ u_X > $