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such that
f
∗
:
f(A)
is an epic arrow, and consequently, we obtain the epic-
monic factorization
f
A
→
f
∗
◦
=
im
f
. For example, in the topos
Set
,
f(A)
={
f(x)
|
A
,
f
∗
(x)
∈
}
:
→
∈
=
x
f(x)
.
Instead, in the weak monoidal topos
DB,
the epic-monic factorization is de-
termined by its categorial symmetry, and the following diagram (see the proof of
Theorem
4
in Sect.
3.2.3
) commutes
A
is the image of the function
f
A
B
, and for each
x
f
=
f
ji
∈
=
f
f
ji
,
such that
T
e
(J(f ))
f
τ
J(f
ji
)
:
A
j
f
ji
|
(f
ji
:
A
j
→
B
i
)
∈
,
τ
J(f)
=
f
and
τ
−
1
,
τ
−
1
J(f)
=
f
J(f
ji
)
:
f
ji
→
B
i
|
(f
ji
:
A
j
→
B
i
)
∈
with
f
τ
J(f)
τ
−
1
J(f
ji
)
, that is, from Definition
28
in Sect.
3.4.1
,
f
≈
τ
J(f
ji
)
≈
τ
−
1
J(f
ji
)
.
In fact, this equivalence is a general result of compositions with monic and epic
arrows, as follows:
The monomorphism τ
−
1
J(f)
:
f
Proposition 65
→
B is the smallest subobject of B
:
→
=
◦
:
→
through which f
A
B factors
.
That is
,
if f
in
g for any g
A
C and a
:
C
→
B then there is a unique k
:
f
→
C making
monic arrow in
commute
,
and hence τ
−
1
J(f)
in
,
as in the ordinary topoi
.
