What is the explicit equivalence relation for an adjunction space $X\cup_f Y$ ?
Intuitively, the construction of $X\cup_f Y$ is pretty straightforward: we begin with the data $(X,Y,B\subseteq Y,\, f:B\rightarrow X)$ and form the adjunction space $X\cup_f Y$ by taking the disjoint union of $X$ and $Y$ then identifying points in $B$ with their image in $X$.
My question is about how to formalize this construction. One often sees notation like '$X/A$' when speaking of quotient spaces, but the definition of a quotient space requires an explicit equivalence relation. With notation as in my first paragraph, the wikipedia article actually defines the notation $Y/B$ to be the adjunct space $X\cup_f Y$ where $X$ is a point.
I would like to be able to write something like $$ X\cup_f Y:= X\coprod Y/\sim, $$ but the equivalence relation I worked out seems a bit, well, messy. What I did manage is the following, but it is not so elegant and I feel I must be overthinking things. Here are my thoughts: set $Z= X\coprod Y$ and define the relation $R\subseteq Z\times Z$ by $R=\cup R_i$ where $$ R_1=\bigg\{\big((x,0),(x,0)\big)\,\,|\,\,x\in X\bigg\} $$ $$ R_2= \bigg\{\big((y,1),(y,1)\big)\,\,|\,\,y\in Y\bigg\} $$ $$ R_3=\bigg\{\big((x,0),(y,1)\big)\,\,|\,\, y\in B \,\, \text{and}\,\, f(y)=x\bigg\} $$ $$ R_4=\bigg\{\big((y,1),(x,0)\big)\,\,|\,\, y\in B \,\, \text{and} \,\, f(y)=x\bigg\}, $$ so then $\alpha\sim \beta\iff (\alpha,\beta)\in R$. With these definitions, I think everything works out but is there a better way to do this?
Your relation still might not be transitive. For example, when $b(y_1)=b(y_2)=x_1$ for $y_1\neq y_2$, you want $(y_0, 1)\sim (y_1, 1)$. Usually you don't write down all the relations of such an equivalence relation and instead give "generating relations". In this case, they just are $$\widetilde R = \left\{ \, \Big((x, 0) , (y, 1)\Big) \,\middle|\, \text{$y\in B$ and $f(y)=x$} \,\right\}.$$ Now you define $R$ to be the smallest equivalence relation containing $\widetilde R$. More precisely, denote by $\mathcal R$ the set of all equivalence relations on $X\coprod Y$ containing $\widetilde R$ and set $$ R = \bigcap_{R'\in\mathcal R} R'. $$ All you need to do is check that arbitrary intersections of equivalence relations are equivalence relations.