A slit band sheet coiling-tension applying device (1; 18; 30; 41; 56) comprising: a first stretched portion (3; 19; 31; 42) configured to have a first cooling roll (6; 21; 33; 44; 57) that is configured to be freely rotatable while having a cylindrical shape; one or more first belts, each having an outer side and an inner side, the outer side being made of a material having a larger friction coefficient than that of a material of the inner side, wherein the one or more first belts are to be brought into contact with the first stretched portion (3; 19; 31; 42) at the inner side of the one or more first belts, and to be stretched in a ring shape to be freely circulated; a first pressing portion (7; 35) configured to be brought into contact with the side of the one or more first belts with the smaller friction coefficient by a predetermined length; a second stretched portion (4; 20; 32; 43) configured to be positioned to face the first stretched portion (3; 19; 31; 42) and to have a second cooling roll (9; 23; 36; 46; 57) that is configured to be freely rotatable while having a cylindrical shape; one or more second belts each having an outer side and an inner side, the outer side being made of a material having a larger friction coefficient than that of a material of the inner side, wherein the one or more second belts are to be brought into contact with the second stretched portion (4; 20; 32; 43) at the inner side of the one or more second belts, and to be stretched in a ring shape to be freely circulated; and a second pressing portion (10; 38) configured to be positioned to face the first pressing portion (7; 35), to be close to the first pressing portion (7; 35), and to be brought into contact with the side of the one or more second belts with the smaller friction coefficient by a predetermined length, characterized in that each of the first cooling roll (21; 44) and the second cooling roll (23; 46) has an inner cylinder (23; 48) on a side of a center shaft and an outer cylinder (24; 49) substantially surrounding the inner cylinder (23; 48), and cooling water (14; 26; 40; 51) is circulated in a space (25; 50) between the inner cylinder (23; 48) and the outer cylinder (24; 49); in that each of the outer cylinders (24; 49) of the first and the second cooling roll (21; 44) has a thickness of 1 to 3 mm; and in that a cross-sectional area of the space is about 2.5 to 5.0 times a cross-sectional area of a pipe on a side of the cooling water entering each cooling roll (21; 44) and a cross-sectional area of a pipe on an outlet side of the cooling water from each cooling roll (21; 44). a first stretched portion (3; 19; 31; 42) configured to have a first cooling roll (6; 21; 33; 44; 57) that is configured to be freely rotatable while having a cylindrical shape; one or more first belts, each having an outer side and an inner side, the outer side being made of a material having a larger friction coefficient than that of a material of the inner side, wherein the one or more first belts are to be brought into contact with the first stretched portion (3; 19; 31; 42) at the inner side of the one or more first belts, and to be stretched in a ring shape to be freely circulated; a first pressing portion (7; 35) configured to be brought into contact with the side of the one or more first belts with the smaller friction coefficient by a predetermined length; a second stretched portion (4; 20; 32; 43) configured to be positioned to face the first stretched portion (3; 19; 31; 42) and to have a second cooling roll (9; 23; 36; 46; 57) that is configured to be freely rotatable while having a cylindrical shape; one or more second belts each having an outer side and an inner side, the outer side being made of a material having a larger friction coefficient than that of a material of the inner side, wherein the one or more second belts are to be brought into contact with the second stretched portion (4; 20; 32; 43) at the inner side of the one or more second belts, and to be stretched in a ring shape to be freely circulated; and a second pressing portion (10; 38) configured to be positioned to face the first pressing portion (7; 35), to be close to the first pressing portion (7; 35), and to be brought into contact with the side of the one or more second belts with the smaller friction coefficient by a predetermined length, characterized in that each of the first cooling roll (21; 44) and the second cooling roll (23; 46) has an inner cylinder (23; 48) on a side of a center shaft and an outer cylinder (24; 49) substantially surrounding the inner cylinder (23; 48), and cooling water (14; 26; 40; 51) is circulated in a space (25; 50) between the inner cylinder (23; 48) and the outer cylinder (24; 49); in that each of the outer cylinders (24; 49) of the first and the second cooling roll (21; 44) has a thickness of 1 to 3 mm; and in that a cross-sectional area of the space is about 2.5 to 5.0 times a cross-sectional area of a pipe on a side of the cooling water entering each cooling roll (21; 44) and a cross-sectional area of a pipe on an outlet side of the cooling water from each cooling roll (21; 44).