let A, B be category; :: thesis: for F being covariant Functor of A,B st F is bijective holds
for G being covariant Functor of B,A st ( for b being Object of B holds F . (G . b) = b ) & ( for a, b being Object of B st <^a,b^> <> {} holds
for f being Morphism of a,b holds F . (G . f) = f ) holds
FunctorStr(# the ObjectMap of G, the MorphMap of G #) = F "
let F be covariant Functor of A,B; :: thesis: ( F is bijective implies for G being covariant Functor of B,A st ( for b being Object of B holds F . (G . b) = b ) & ( for a, b being Object of B st <^a,b^> <> {} holds
for f being Morphism of a,b holds F . (G . f) = f ) holds
FunctorStr(# the ObjectMap of G, the MorphMap of G #) = F " )
assume A1: F is bijective ; :: thesis: for G being covariant Functor of B,A st ( for b being Object of B holds F . (G . b) = b ) & ( for a, b being Object of B st <^a,b^> <> {} holds
for f being Morphism of a,b holds F . (G . f) = f ) holds
FunctorStr(# the ObjectMap of G, the MorphMap of G #) = F "
let G be covariant Functor of B,A; :: thesis: ( ( for b being Object of B holds F . (G . b) = b ) & ( for a, b being Object of B st <^a,b^> <> {} holds
for f being Morphism of a,b holds F . (G . f) = f ) implies FunctorStr(# the ObjectMap of G, the MorphMap of G #) = F " )
assume that
A2: for b being Object of B holds F . (G . b) = b and
A3: for a, b being Object of B st <^a,b^> <> {} holds
for f being Morphism of a,b holds F . (G . f) = f ; :: thesis: FunctorStr(# the ObjectMap of G, the MorphMap of G #) = F "
hence FunctorStr(# the ObjectMap of G, the MorphMap of G #) = F " by A1, A4, Th4, YELLOW18:1; :: thesis: verum