On p.294 of Conceptual Mathematics 2nd ed., Exercise 12 states:
For a given map $h: Y \rightarrow Z$, consider all parallel pairs $f, g: X \rightarrow Y$ (for various $X$) such that $hf = hg$. Formulate the notion of a universal such; call it $X_h$. Show that $X_h \rightrightarrows Y$ is reflexive, symmetric, transitive, and jointly monomorphic. Here 'reflexive' you know from our discussion of directed graphs, 'symmetric' means there is an involution $\sigma$ of $X_h$ whose right action interchanges the universal $f$ and $g$. 'Jointly monomorphic' means the map $X \rightarrow Y \times Y$ with label $<f, g>$ is injective. 'Transitivity' involves a trio of test maps $T \rightarrow Y$. (An STM reflexive graph is called an equivalence relation on $Y$; in many categories every equivalence relation arises as the universal $X_h$ for some $h$.)
I am confused about the definitions given for 'symmetric' and 'transitivity.'
My understanding of the universality of the premise is given another object $A$ with maps $i,j: A \rightarrow Y$ such that $hi = hj$, there is a unique map $x: A \rightarrow X$ such that $fx = i$ and $gx = j$.
For the given definition of symmetry I am confused as to what the "involution's right action interchanging $f$ and $g$ means. I know an involution $\sigma: X_h \rightarrow X_h$ means $\sigma\sigma = 1_{X_h}$. I think a right action of $X$ on $Y$ is a map $X \times Y \rightarrow Y$. I am not sure how an involution can have a right action. I also don't know what it means to "interchange" $f$ and $g$.
The "definition" of transitivity is even more confusing.. what is a "trio of test maps" ?
Right action presumably means pre-composition, so what it is saying is if $p,q : X_h \rightrightarrows Y$ are the universal parallel arrows, then $p\circ\sigma = q$ and $q\circ\sigma = p$.
For transitivity, the idea is that you have three maps $i,j,k : T \to Y$ and you consider the three pairs that induces $i,j : T \rightrightarrows Y$, $j,k : T \rightrightarrows Y$ and $i,k : T \rightrightarrows Y$ and how they relate, which I'll leave to you. Basically, you want to prove a categorical reformulation of the statement: if $(i,j)\in X_h \subseteq Y\times Y$ and $(j,k)\in X_h$ then $(i,k)\in X_h$.