Pullback

Remarks

A pullback is the limit of the diagram $X \xrightarrow{f} Z \xleftarrow{g} Y$, so it is unique up to unique isomorphism. If the product $X \times Y$ exists, the pullback is the equalizer of $f \circ \pi_1$ and $g \circ \pi_2 \colon X \times Y \to Z$.

Example: in the category of sets, let $P = \set{(x, y) \in X \times Y : f(x) = g(y)}$ with the two coordinate maps. Clause 1 holds by the choice of $P$. Given $q_1, q_2$ with $f \circ q_1 = g \circ q_2$, the map $u(w) = (q_1(w), q_2(w))$ lands in $P$ and is the only map with the required coordinates. When $g$ is the inclusion of a subset $Y \subseteq Z$, the pullback is the preimage $f^{-1}(Y)$ with its inclusion into $X$.

Non-example: in the category of sets, let $X = Y = \{0, 1\}$, $Z = \{*\}$ and let $f, g$ be the constant maps. The set $\{0, 1\}$ with $p_1 = p_2 = \id$ satisfies clause 1, but the pair $q_1 = \id$, $q_2 = (0 \mapsto 1, 1 \mapsto 0)$ on $W = \{0, 1\}$ admits no $u$ with $p_1 \circ u = q_1$ and $p_2 \circ u = q_2$, so clause 2 fails.