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∩
is
−
A
=
asimplearrow
f
,
Tf
is
−
A
=
is
B
∩
f
∩
f
=
is
B
◦
f
◦
TB
∩
TA
=
(from
Proposition
7
if
f
is simple arrow then
f
=
f
.If
f
is a complex arrow
with
A
=
1
≤
j
≤
m
A
j
and
B
=
1
≤
i
≤
k
B
i
then
is
−
A
=
1
≤
j
≤
m
is
−
1
⊆
TA
∩
TB
)
A
j
:
TA
j
→
A
j
and
is
B
=
1
≤
i
≤
k
is
B
i
:
is
−
1
A
with
is
−
1
id
A
A
j
={
id
R
:
R
→
R
|
R
∈
A
j
}
, so that
=
is
B
id
B
B
i
→
TB
i
with
is
B
i
={
id
R
:
R
→
R
|
R
∈
B
i
}
and hence
=
. Consequently,
,
f
ji
=
f
ji
so
Tf
f
for each
f
ji
=
is
−
1
Tf
ji
=
is
B
i
◦
f
ji
◦
A
j
∈
with
f
ji
:
A
j
→
B
i
∈
Tf
f
that, from Definition
22
,
Tf
f
.
A
with
id
A
=
For each identity arrow
id
A
:
A
→
TA
, we obtain the identity
TA
with
id
TA
=
T id
A
=
id
A
=
arrow
id
TA
=
T id
A
:
→
TA
TA
. For the composi-
is
−
A
=
tion of any two arrows
f
:
A
→
B
and
g
:
B
→
C
,
T(g
◦
f)
=
is
C
◦
(g
◦
f)
◦
is
−
A
=
(is
−
B
◦
is
−
A
=
is
C
◦
g
◦
id
B
◦
f
◦
is
C
◦
g
◦
is
B
)
◦
f
◦
Tg
◦
Tf
. Thus, from
Definition
23
,
T(g
◦
f)
=
Tg
◦
Tf
. Consequently, the power-view operator
T
is a
DB
endofunctor.
Thus,
T(Tf)
is
−
A
)
T is
−
A
=
=
T(is
B
◦
f
◦
=
T is
B
◦
Tf
◦
(from Example
21
bel-
low)
Tf
. It is easy to verify that
T
is a 2-endofunctor and
that preserves the properties of arrows: if
f
is an isomorphism then
Tf
is a compo-
sition of three isomorphism and hence an isomorphism as well. If
f
is monic then
Tf
=
id
TB
◦
Tf
◦
id
TA
=
is
−
A
, that is, a composition of three monic arrows (each isomorphic
arrow is monic and epic as well) and hence is a monomorphism as well. The proof
when
f
is epic is similar. The functorial properties for different cases of complex
arrows are presented with details in point 2 of Corollary
12
in Sect.
3.2.4
.
=
is
B
◦
f
◦
Let us show that the morphism
Tf
:
TA
→
TB
, for a given morphisms
f
=
α
∗
(MakeOperads(
M
AB
))
with
M
AB
={
Ψ
}
,
M
AB
=
MakeOperads(
M
AB
)
=
{
S
B
, corresponds to the definition
of arrows in
DB
category, specified by Theorem
1
, that is, there exists an SOtgd
Φ
of a schema mapping such that
Tf
q
1
,...,q
k
,
1
r
∅
}
and
S
={
∂
1
(q
i
)
|
1
≤
i
≤
k
}⊆
α
∗
(MakeOperads(
=
{
}
))
.
We will show that
Φ
is just the SOtgd of the schema mapping
Φ
M
AB
:
A
→
B
,
both with a mapping-interpretation
α
, used in definition of the morphism
f
.
From Theorem
2
above we obtain
is
−
1
A
Tf
=
is
B
◦
f
◦
=
α
∗
MakeOperads
{
Φ
B
}
◦
α
∗
(
M
AB
)
◦
α
∗
MakeOperads(Φ
A
)
=
α
∗
MakeOperads
{
Φ
B
}
◦
MakeOperads(Φ
A
)
M
AB
◦
α
∗
{
1
r
∅
}
◦
M
AB
◦
{
1
r
∅
}
=
1
r
|
r
∈
S
B
}∪{
1
r
|
r
∈
S
A
}∪{
α
∗
{
1
r
∅
}
◦
M
AB
=
1
r
|
∈
S
B
}∪{
r
α
∗
{
1
r
∅
}
◦{
q
1
,...,q
k
,
1
r
∅
}
=
1
r
|
∈
S
B
}∪{
r
α
∗
{
1
r
∅
}
◦{
q
1
,...,q
k
,
1
r
∅
}
=
1
r
|
r/
∈
S
}∪{
1
r
|
r
∈
S
}∪{
α
∗
{
1
r
∅
}
◦{
q
1
,...,q
k
,
1
r
∅
}
=
1
r
|
r
∈
S
B
}∪{
α
∗
{
q
1
,...,q
k
,
1
r
∅
}
=
α
∗
MakeOperads
{
}
,
=
1
r
|
r/
∈
S
}∪{
Φ
