let A, B be category; :: thesis: for F being contravariant Functor of A,B st F is bijective holds
for G being contravariant Functor of B,A st ( for a being object of A holds G . (F . a) = a ) & ( for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds G . (F . f) = f ) holds
FunctorStr(# the ObjectMap of G,the MorphMap of G #) = F "
let F be contravariant Functor of A,B; :: thesis: ( F is bijective implies for G being contravariant Functor of B,A st ( for a being object of A holds G . (F . a) = a ) & ( for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds G . (F . f) = f ) holds
FunctorStr(# the ObjectMap of G,the MorphMap of G #) = F " )
assume A1: F is bijective ; :: thesis: for G being contravariant Functor of B,A st ( for a being object of A holds G . (F . a) = a ) & ( for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds G . (F . f) = f ) holds
FunctorStr(# the ObjectMap of G,the MorphMap of G #) = F "
let G be contravariant Functor of B,A; :: thesis: ( ( for a being object of A holds G . (F . a) = a ) & ( for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds G . (F . f) = f ) implies FunctorStr(# the ObjectMap of G,the MorphMap of G #) = F " )
assume that
A2: for b being object of A holds G . (F . b) = b and
A3: for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds G . (F . f) = f ; :: thesis: FunctorStr(# the ObjectMap of G,the MorphMap of G #) = F "

A4: now

let b be object of A; :: thesis: (G * F) . b = (id A) . b
thus (G * F) . b = G . (F . b) by FUNCTOR0:34
.= b by A2
.= (id A) . b by FUNCTOR0:30 ; :: thesis: verum

end;

now

let a, b be object of A; :: thesis: ( <^a,b^> <> {} implies for f being Morphism of a,b holds (G * F) . f = (id A) . f )
assume A5: <^a,b^> <> {} ; :: thesis: for f being Morphism of a,b holds (G * F) . f = (id A) . f
let f be Morphism of a,b; :: thesis: (G * F) . f = (id A) . f
thus (G * F) . f = G . (F . f) by A5, FUNCTOR3:7
.= f by A3, A5
.= (id A) . f by A5, FUNCTOR0:32 ; :: thesis: verum

end;

then G * F = id A by A4, YELLOW18:1;
hence FunctorStr(# the ObjectMap of G,the MorphMap of G #) = F " by A1, Th5; :: thesis: verum