Catalyzes the formation of a Delta12 double bond, acting on monounsaturated fatty acyl subtrates like palmitoleoyl-CoA ((9Z)-hexadecenoyl-CoA) and oleoyl-CoA ((9Z)-octadecenoyl-CoA) with higher desaturation activity on (9Z)-octadecenoyl-CoA than (9Z)-hexadecenoyl-CoA. Requires preexisting cis double bond at the Delta9 position of fatty acyls to be able to insert the Delta12 double bond. Delta12-desaturation of (9Z)-octadecenoyl-CoA in insects produces (9Z,12Z)-octadecadienoyl-CoA (linoleoyl-CoA) which may be used to supply precursors of crucial mediators of immunity and reproduction and other essential functions. (9Z)-octadecenoyl-CoA + 2 Fe(II)-[cytochrome b5] + 2 H(+) + O2 = (9Z,12Z)-octadecadienoyl-CoA + 2 Fe(III)-[cytochrome b5] + 2 H2O (9Z)-hexadecenoyl-CoA + 2 Fe(II)-[cytochrome b5] + 2 H(+) + O2 = (9Z,12Z)-hexadecadienoyl-CoA + 2 Fe(III)-[cytochrome b5] + 2 H2O Expected to bind 2 Fe(2+) ions per subunit. The histidine box domains may contain the active site and/or be involved in metal ion binding. Belongs to the fatty acid desaturase type 1 family.