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In fact, for any other cocone diagram
(ν
,D)
with a vertex
D
and with natu-
ral transformation
ν
:
D
that provides the components (ptp arrows)
h
i
=
F
→
ν
(B
i
)
ν
B
i
, there is a unique arrow
k
D
such that
h
i
=
=
:
ColimF
→
k
◦
h
i
,
1
≤
n
.
This colimit diagram is then translated by the functor
H
=
i
≤
DB
(Υ,
_
)
in an
analogous colimit diagram in
Set
with the cocone-base
HF
, as presented by the
right-hand side of the diagram above for any fixed cocone monic arrow
h
i
:
→
ColimF
. Let us show that for any other cocone
E
in
Set
, for the same cocone-base
HF
(where
H
B
i
Set
) there is a unique arrow (a function)
k
from
DB
(Υ, ColimF)
to the set
E
(vertex of a cocone
E
) such that the diagram com-
posed of these two cocones commutes. That is, for each object
DB
(Υ,B
i
)
in the
cocone-base
HF
, we obtain a function
l
i
which is a component of cocone
E
in
Set
:
=
DB
(Υ,
_
)
:
DB
→
l
i
=
k
◦
DB
(Υ,h
i
)
:
DB
(Υ,B
i
)
→
E,
(8.2)
for each 1
n
.
From the functorial property of
H
≤
i
≤
=
DB
(Υ,
_
)
:
DB
→
Set
, we have that for
the function
DB
(Υ,h
i
)
:
DB
(Υ,B
i
)
→
DB
(Υ, ColimF)
it holds that for each DB-
morphism
(g
:
Υ
→
B
i
)
∈
DB
(Υ,B
i
),
DB
(Υ,h
i
)(g)
=
h
i
◦
g
is a DB-morphism
with the set of ptp (simple) arrows
h
ij
◦
g
l
|
h
i
◦
g
=
g
h
i
cod(g
l
)
=
dom(h
ij
),g
l
∈
,h
ij
∈
where for each monic simple arrow
h
ij
,
h
ij
=
T(dom(h
ij
))
and
T
dom(g
l
)
∩
T
cod(g
l
)
⊆
T
cod(g
l
)
=
T
dom(h
ij
)
=
h
ij
.
g
l
⊆
That is,
h
ij
◦
g
l
=
h
ij
∩
g
l
=
g
l
and hence, by Definition
28
in Sect.
3.4.1
, these two
simple morphisms are equivalent arrows in
DB
(up to isomorphism),
h
ij
◦
g
l
≈
g
l
.
h
i
◦
g
g
Consequently,
≈
, i.e.,
DB
(Υ,h
i
)(g)
=
h
i
◦
g
≈
g
, and from (
8.2
)
l
i
(g)
=
=
k(
DB
(Υ,B
i
)(g))
k(g)
.
Thus, from (
8.2
),
∀
g
∈
DB
(Υ,B
i
).k(g)
=
l
i
(g)
, for each 1
≤
i
≤
n
, that is, the
function
k
is uniquely defined by the set of cocone functions
l
i
,
1
n
.
Let us consider this problem alternatively by category symmetry, where each
DB-morphism
f
is substituted by its dual object, equal to the information flux
f
: Each hom-set
DB
(Υ,B
i
)
in
Set
is isomorphic in
Set
to the complete sublat-
tice
(
≤
i
≤
{
f
|
f
{
f
|
f
⊗
B
i
}
=
TB
i
}
:
−→
Υ
,
)
(
,
)
(because each arrow
f
Υ
B
i
corresponds to the closed object
f
Υ
⊗
B
i
=
TB
i
). On the other hand,
H ColimF
=
DB
(Υ, ColimF)
is isomorphic to the complete sublattice
(S,
)
, where
B
i
∈
F
TB
i
=
B
i
∈
F
TB
i
}
={
f
|
f
S
. Hence, all arrows of the cocone
Hν
,
{
f
|
f
DB
(Υ,h
i
)
:
DB
(Υ,B
i
)
→
DB
(Υ, ColimF)
are injective functions
(
TB
i
}
,
)
(also all arrows in the base diagram
HF
in
Set
are injective functions
of the form
(
{
f
|
f
TB
j
}
,
)
⊆
(
{
f
|
f
TB
k
}
,
)
).
)
⊆
(S,
