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categorical pullback
Pullbacks
Let $f:X\to B$ and $g:Y\to B$ be morphisms in a category $\mathcal{C}$. Then a pullback diagram, or pullback square of $f$ and $g$ is a commutative diagram
$\xymatrix{A\ar[d]_{p}\ar[r]^{q}&Y\ar[d]^{g}\\ X\ar[r]_{f}&B.}$ 
such that if we have another commutative diagram
$\xymatrix{Z\ar[d]_{r}\ar[r]^{s}&Y\ar[d]^{g}\\ X\ar[r]_{f}&B.}$ 
then there is a unique morphism $h:Z\to A$ with the commutative diagram
$\xymatrix{Z\ar@/^{1}ex/[rrd]^{s}\ar@/_{1}ex/[rdd]_{r}\ar@{.>}[rd]h&&\\ &A\ar[d]_{p}\ar[r]^{q}&Y\ar[d]^{g}\\ &X\ar[r]_{f}&B.}$ 
A pullback of $(f,g)$ is the ordered triple $(A,q,p)$. We also say that $p$ is a pullback of $g$ along $f$, and $q$ a pullback of $f$ along $g$. When a pullback of $(f,g)$ exists, it is unique up to isomorphism. The object $A$, and the morphisms $p$ and $q$ in the diagram above are often denoted by
$X\times_{B}Y,\qquad 1_{X}\times_{B}g\qquad\mbox{and}\qquad f\times_{B}1_{Y}$ 
respectively, and the uniquely determined morphism $h$ by
$\displaystyle{r\choose s}.$ 
It is easy to see that
$X\times_{B}Y\cong Y\times_{B}X,$ 
whenever one (and hence the other) exists.
Remarks.

The pullback of $f$ and $g$ can be equivalently defined as a limiting cone over the diagram $X\to B\leftarrow Y$. In other words, a pullback diagram is a terminal object in the category of commutative squares of the form
$\xymatrix{Z\ar[d]\ar[r]&Y\ar[d]^{g}\\ X\ar[r]_{f}&B.}$ 
A category $\mathcal{C}$ is said to have pullbacks if every diagram $X\to B\leftarrow Y$ can be completed into a pullback diagram.

The notion of pullbacks can be generalized: let $\{x_{i}:C_{i}\to C\mid i\in I\}$ be a collection of morphisms indexed by set $I$, considered as a small diagram. The generalized pullback of the $x_{i}$’s is just the limiting cone of the diagram. Using this definition, the generalized pullback of one morphism $f$ is the identity morphism of $\operatorname{dom}(f)$, the domain of $f$, and the generalized pullback of the empty set is a terminal object.

A pullback is sometimes known as an amalgamated sum.
Pushouts
Dually, given morphisms $f:B\to X$ and $g:B\to Y$, a pushout square of $f$ and $g$ is a commutative diagram
$\xymatrix{B\ar[d]_{f}\ar[r]^{g}&Y\ar[d]^{s}\\ X\ar[r]_{t}&A}$ 
such that if we have another commutative diagram
$\xymatrix{B\ar[d]_{f}\ar[r]^{g}&Y\ar[d]^{u}\\ X\ar[r]_{v}&Z}$ 
Then there is a unique morphism $h:A\to Z$ with the following commutative diagram:
$\xymatrix{B\ar[d]_{f}\ar[r]^{g}&Y\ar[d]^{s}\ar@/^{1}ex/[ddr]^{u}&\\ X\ar[r]_{t}\ar@/_{1}ex/[drr]_{v}&A\ar@{.>}[dr]h&\\ &&Z.}$ 
The pair $(s,t)$ of morphisms is a pushout of $(f,g)$. $s$ is the the pushout of $f$ along $g$, and $t$ the pushout of $g$ along $f$. Like pullbacks, pushouts are unique up to unique isomorphism when they exist. The object $A$ and the morphisms $s$ and $t$ are typically written
$X\amalg_{B}Y,\qquad f\amalg_{B}1_{Y}\qquad\mbox{and}\qquad 1_{X}\amalg_{B}g,$ 
and the unique morphism $h$ is denoted by
$(f\enspace g).$ 
Remark. The pushout of $f$ and $g$ can be thought of as the limiting cocone under the diagram $X\leftarrow B\to Y$. Equivalently, they are initial objects in the category of commutative squares whose top edge is $B\to Y$ and left edge is $B\to X$. A category is said to have pushouts if every diagram $X\leftarrow B\to Y$ can be completed to a pushout diagram. The generalized pushout is defined as the limiting cocone under the diagram consisting of morphisms $y_{i}:B\to B_{i}$, where $i$ belongs to some set $I$.
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