Accepted answer
15 ÷ 5 = 3 mg/mL. Concentration is vial content divided by diluent volume, so 15 mg of peptide in 5 mL of phosphate-buffered diluent gives 3 mg/mL. On a U-100 barrel one unit is 0.01 mL, so one unit of this solution carries 0.03 mg — 30 µg. That is the number to write on the label, because you will not reconstruct it from memory at an awkward moment.
Add the diluent down the vial wall rather than directly onto the cake. Peptides are surface-active and shear at an air–liquid interface, so a jet of water into a lyophilised puck generates foam, and foam is aggregated protein at the interface, not just air.
Photograph the vial against a matte black card with a single point light source off to one side, not with a flash from the front.
Room temperature before drawing is worth the ten minutes. Cold solution is more viscous, draws slower, and is more likely to pull a bubble past the plunger seal.
Published data on syringe dead space in the context of injection-equipment programmes quantifies low-dead-space designs as retaining under 2 µL against 35 µL or more.
The caveat is that this is not a recommendation to administer anything. Research-use-only material is not approved for human use.
The practical summary: fine gauge, gentle swirl, diluent down the wall, room temperature before drawing, and check the syringe scale.