Some definitions of interest.
compose	Def  (f o g)(x) == f(g(x))
	Thm*  A,B,C:Type, f:(BC), g:(AB). f o g  AC
	Thm*  A,B,C:Type, f:(B inj C), g:(A inj B). f o g  A inj C
	Thm*  f:(B onto C), g:(A onto B). f o g  A onto C
injection_type	Def  A inj B == {f:(AB)| Inj(A; B; f) }
	Thm*  A,B:Type. A inj B  Type
inject	Def  Inj(A; B; f) == a1,a2:A. f(a1) = f(a2)  B  a1 = a2
	Thm*  A,B:Type, f:(AB). Inj(A; B; f)  Prop

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