two theorems without proofs: every equalizer is monomorphism; every coequalizer is epimorphisms

examples/category-theory/Equalizer.gf

@@ -1,4 +1,4 @@
-abstract Equalizer = Categories ** {
+abstract Equalizer = Morphisms ** {
 
 cat Equalizer ({c} : Category) ({x,y,z} : El c) (f,g : Arrow x y) (e : Arrow z x) ;
 
@@ -15,13 +15,20 @@ fun idEqualizer : ({c} : Category)
                -> Equalizer f f (id x) ;
 def idEqualizer {c} {x} {y} f = equalizer f f (id x) (eqCompL f (eqRefl (id x))) ;
 
+fun equalizer2mono :  ({c} : Category)
+                   -> ({x,y,z} : El c)
+                   -> (f,g : Arrow x y)
+                   -> (e : Arrow z x)
+                   -> Equalizer f g e
+                   -> Mono e ;
+-- def equalizer2mono = ...
 
 cat CoEqualizer ({c} : Category) ({x,y,z} : El c) (e : Arrow y z) (f,g : Arrow x y) ;
 
 data coequalizer :  ({c} : Category)
                  -> ({x,y,z} : El c)
-                 -> (f,g : Arrow x y)
                  -> (e : Arrow y z)
+                 -> (f,g : Arrow x y)
                  -> EqAr (comp e f) (comp e g)
                  -> CoEqualizer e f g ;
 
@@ -29,6 +36,14 @@ fun idCoEqualizer : ({c} : Category)
                  -> ({x,y} : El c)
                  -> (f : Arrow x y)
                  -> CoEqualizer (id y) f f ;
-def idCoEqualizer {c} {x} {y} f = coequalizer f f (id y) (eqCompR (eqRefl (id y)) f) ;
+def idCoEqualizer {c} {x} {y} f = coequalizer (id y) f f (eqCompR (eqRefl (id y)) f) ;
+
+fun coequalizer2epi :  ({c} : Category)
+                    -> ({x,y,z} : El c)
+                    -> ({e} : Arrow y z)
+                    -> ({f,g} : Arrow x y)
+                    -> CoEqualizer e f g
+                    -> Epi e ;
+-- def coequalizer2epi = ...
 
 }