This study will include 2 groups (cohorts) of participants:

In Cohort A, this study will look at the amount of LY2605541 that is present in the body after it is injected in 3 different locations in the body of the same participant. At least 16 days will pass between each injection.

In Cohort B, the study will look at how the body absorbs, distributes, and disposes of LY2605541 in participants aged 65 and over.

Participants may enroll in only one cohort.

Inclusion Criteria:

• Cohort A: are overtly healthy male or female participants, as determined by medical history and physical examination, aged ≥18 to ≤55 years of age
• Cohort B: are male or female elderly (≥65 years of age) participants that are overtly healthy, or have stable, chronic medical conditions (including Type 2 Diabetes Mellitus [T2DM]) that, in the investigator's opinion, will not significantly alter the disposition of the drug, will not place the participant at increased risk by participating in the study, and will not interfere with interpretation of the data
• Male participants agree to use a reliable method of birth control during the study and for 3 months following the last dose of the investigational product
• Female participants are women of child-bearing potential who test negative for pregnancy at the time of screening based on a urine pregnancy test and agree to use a reliable method of birth control during the study and for 1 month following the last dose of the investigational product
• Female participants are post-menopausal women, or women not of child-bearing potential due to surgical sterilization (at least 3 months after surgical hysterectomy, or at least 3 months after bilateral oophorectomy or bilateral tubal ligation/occlusion with or without hysterectomy) confirmed by medical history, or menopause
• Have a body mass index (BMI) of 18.5 to 40 kilogram per meter square (kg/m^2), inclusive, at time of admission for Period 1
• Have given written informed consent approved by Lilly and the ethical review board (ERB) governing the site

Exclusion Criteria:

• Are participants that intend to start a change in diet or lose weight during the study; participants will be expected to maintain their approximate current weight throughout the study
• Are excessive smokers (>10 cigarettes per day); participants who smoke ≤10 cigarettes per day must be able to stop smoking while residing at the clinical research unit (CRU)

Healthy Participants - Cohort A

• Have Type 1 Diabetes Mellitus (T1DM), Type 2 Diabetes Mellitus (T2DM), or a fasting blood glucose (BG) >110 milligrams per deciliter (mg/dL) (6.1 millimoles per liter [mmol/L]) with a glycated hemoglobin (HbA1c) of >6.2%
• Intend to use over-the-counter medication within the 7 days or prescription medication within 14 days prior to dosing (apart from contraceptive medication, vitamin/mineral supplements, occasional paracetamol, thyroxine replacement therapy, and hormone replacement therapy). Antihypertensive and lipid lowering agents are permitted, but must have been administered at stable dose(s) for at least 1 month.

Elderly Participants - Cohort B

• Have T1DM
• Are T2DM participants that require insulin treatment, or have a HbA1c of >10%
• Have started new chronic medication(s) or changed the dose of an existing chronic medication <1 month prior to dosing. Participants are permitted to continue ongoing chronic medications (for example antihypertensive agents, aspirin, non-steroidal anti-inflammatory drugs, lipid lowering agents, thyroxine, hormone replacement therapy, and/or oral anti-diabetic if participants have T2DM). Other stable chronic conditions may be permitted if deemed acceptable by the investigators and Lilly clinical pharmacologist or clinical research physician.

All enrolled participants.