let A, B be non empty transitive with_units AltCatStr ; :: thesis: for F, G being covariant Functor of A,B
for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds ((idt F) ! b) * (F . f) = (F . f) * ((idt F) ! a)
let F, G be covariant Functor of A,B; :: thesis: for a, b being object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds ((idt F) ! b) * (F . f) = (F . f) * ((idt F) ! a)
let a, b be object of A; :: thesis: ( <^a,b^> <> {} implies for f being Morphism of a,b holds ((idt F) ! b) * (F . f) = (F . f) * ((idt F) ! a) )
assume A1: <^a,b^> <> {} ; :: thesis: for f being Morphism of a,b holds ((idt F) ! b) * (F . f) = (F . f) * ((idt F) ! a)
A2: ( <^a,a^> <> {} & <^b,b^> <> {} ) by ALTCAT_2:def 7;
let f be Morphism of a,b; :: thesis: ((idt F) ! b) * (F . f) = (F . f) * ((idt F) ! a)
thus ((idt F) ! b) * (F . f) = (idm (F . b)) * (F . f) by Th6
.= (F . (idm b)) * (F . f) by Th2
.= F . ((idm b) * f) by A1, A2, FUNCTOR0:def 24
.= F . f by A1, ALTCAT_1:24
.= F . (f * (idm a)) by A1, ALTCAT_1:def 19
.= (F . f) * (F . (idm a)) by A1, A2, FUNCTOR0:def 24
.= (F . f) * (idm (F . a)) by Th2
.= (F . f) * ((idt F) ! a) by Th6 ; :: thesis: verum