An H-magmoid $A$ consists of the following.
A type $A_0$, whose elements are called objects. Typically $A$ is coerced to $A_0$ in order to write $x:A$ for $x:A_0$.
For each $a,b:A$, a type $hom_A(a,b)$, whose elements are called arrows or morphisms.
For each $a,b,c:A$, a function
called composition, and denoted infix by $g \mapsto f \mapsto g \circ f$, or sometimes $gf$.