We study Bridgeland moduli spaces of semistable objects of $$(-1)$$
(
-
1
)
-classes and $$(-4)$$
(
-
4
)
-classes in the Kuznetsov components on index one prime Fano threefold $$X_{4d+2}$$
X
4
d
+
2
of degree $$4d+2$$
4
d
+
2
and index two prime Fano threefold $$Y_d$$
Y
d
of degree d for $$d=3,4,5$$
d
=
3
,
4
,
5
. For every Serre-invariant stability condition on the Kuznetsov components, we show that the moduli spaces of stable objects of $$(-1)$$
(
-
1
)
-classes on $$X_{4d+2}$$
X
4
d
+
2
and $$Y_d$$
Y
d
are isomorphic. We show that moduli spaces of stable objects of $$(-1)$$
(
-
1
)
-classes on $$X_{14}$$
X
14
are realized by Fano surface $$\mathcal {C}(X)$$
C
(
X
)
of conics, moduli spaces of semistable sheaves $$M_X(2,1,6)$$
M
X
(
2
,
1
,
6
)
and $$M_X(2,-1,6)$$
M
X
(
2
,
-
1
,
6
)
and the correspondent moduli spaces on cubic threefold $$Y_3$$
Y
3
are realized by moduli spaces of stable vector bundles $$M^b_Y(2,1,2)$$
M
Y
b
(
2
,
1
,
2
)
and $$M^b_Y(2,-1,2)$$
M
Y
b
(
2
,
-
1
,
2
)
. We show that moduli spaces of semistable objects of $$(-4)$$
(
-
4
)
-classes on $$Y_{d}$$
Y
d
are isomorphic to the moduli spaces of instanton sheaves $$M^{inst}_Y$$
M
Y
inst
when $$d\ne 1,2$$
d
≠
1
,
2
, and show that there are open immersions of $$M^{inst}_Y$$
M
Y
inst
into moduli spaces of semistable objects of $$(-4)$$
(
-
4
)
-classes when $$d=1,2$$
d
=
1
,
2
. Finally, when $$d=3,4,5$$
d
=
3
,
4
,
5
we show that these moduli spaces are all isomorphic to $$M^{ss}_X(2,0,4)$$
M
X
ss
(
2
,
0
,
4
)
.