The Properties of DB Category - Big Data Integration Theory - page 374

Thus,

f ⊗ g ◦ g = T f ◦ f ∩ T g ◦ g = T f ∩ f ∩ T g ∩ g f ◦

= f ∩ f ∩ g ∩ g = f ∩ g ∩ (f ∩ g)

T f ∩ g ∩

T(f = f ⊗ g ∩ f = ∩ g) ⊗ g

f ⊗

g ◦ = (f ⊗ g),

so that from Definition 23 ,
f ◦

f ⊗ g ◦ g = f ⊗ g ◦

(f ⊗ g). (8.1)

Let us show that this property of composition is valid for complex morphisms f :

C , f : D and g :

A → C → E , g : B → B → F as well. Then,

(f ⊗ g )

◦ (f ⊗ g)

f i ⊗

f ⊗ g

g j ◦

f ⊗ g

f i ⊗ g j ∈ = (f i ⊗ g j ) | ,f i ⊗ g j ∈ ,

g j

dom f i ⊗

cod(f i ⊗ g j ) =

f i ⊗

f ⊗ g

g j ◦

f ⊗ g

f i ⊗ g j ∈

= (f i ⊗ g j ) | ,f i ⊗ g j ∈ ,

dom g j dom f i , cod(g j )

= =

cod(f i )

f i ⊗

f ◦ g ◦

g j ◦

f g j

f i ◦ ,g j ◦ = (f i ⊗ g j ) | f i ∈ g j ∈

from the fact that all f i ,f i ,g j ,g j are simple arrows and ( 8.1 )

f i ◦

f ◦ g ◦

f i ⊗ g j ◦ g j |

f g j

f i ◦ ,g j ◦ = f i ∈ g j ∈

id R |

f ◦ g ◦

g j |

f g j

∈ f i ◦ f i ∩ g j ◦

f i ◦ ,g j ◦ = R f i ∈ g j ∈

(by Definition 22 )

(f ◦ (g ◦

f) ⊗ g)

= .

Consequently, (f ◦ (g ◦ (f ⊗ g ) f) ⊗ g) = ◦ (f ⊗ g) , and hence the compositional

property for functor ⊗ is satisfied.