I'm reading the book "Category Theory for the Sciences" by David Spivak and on page 87, in a section about pushouts, I was faced with the following problem:
Let $i: R \subseteq X \times X$ be an equivalence relation (...). Composing with the projections $\pi_1, \pi_2: X \times X \to X$, we have two maps, $\pi_1 \circ i: R \to X$ and $\pi_2 \circ i: R \to X$.
(a) ...
(b) If $i: R \subseteq X \times X$ is not assumed to be an equivalence relation, we can still define this pushout. Is there a relationship between the pushout $X \overset{\pi_1 \circ i}{\longleftarrow} R \overset{\pi_2 \circ i}{\longrightarrow} X$ and the equivalence relation generated by $R \subseteq X \times X$?
To which the author's answer is:
Yes, $X \sqcup_R X$ is isomorphic to the quotient $X/\sim$, where $\sim$ is the equivalence relation generated by R.
As an example, let $R := \emptyset$ and $X := \{1,2,3\}$.
Since $R$ is the empty set, then the pushout $X \sqcup_R X $ is isomorphic to $X \sqcup X$. Intuition tells me the disjoint union $X \sqcup X$ should contain every element twice, so its cardinality is 6.
However, the equivalence relation $\sim$ generated by $R$ is the trivial relation $\sim = \{(1,1), (2,2), (3,3)\}$, and the quotient $X/\sim = \{\{1\}, \{2\}, \{3\}\}$, with cardinality 3.
So, how can $X \sqcup_R X $ be isomorphic to $X/\sim$?
The claim seems to only be true if $R$ contains the diagonal $\Delta\subset X\times X$, because then the pushout of $$X\leftarrow R\rightarrow X$$ will identify (or "glue") together the two copies of $X$ to make a single copy of $X$, which will then be further quotiented by the equivalence relation generated by $R$. When $R$ does not contain the diagonal, as in the example you found, one is left with parts of the two copies of $X$ that are not identified.
More formally, recall that $X\sqcup_R X\cong \color{red}{X}\sqcup R\sqcup \color{blue}{X}/{\sim}$ where $\sim$ is the equivalence relation generated by $$(a,b)\sim \color{red}{a},\qquad (a,b)\sim \color{blue}{b}\qquad \text{ for all }(a,b)\in R$$ where the red and blue are used to distinguish the copies of $X$. If $\Delta\subseteq R$, then $\color{red}{a}\sim \color{blue}{a}$ for all $a\in X$, and every element of $X$ is related to some element of $R$, so that every equivalence class of $\color{red}{X}\sqcup R\sqcup \color{blue}{X}/{\sim}$ has a representative in $\color{red}{X}$, and therefore $X\sqcup_R X$ is just a single copy of $X$ modulo the equivalence relation $\approx$ generated by $$a\approx b\qquad\text{ for all }(a,b)\in R$$