5Step 1. Observe that ΣX is the colimit of any diagram of the form 0←X→X←X→…←X→0, and
ΣX ⊔ ΣX ⊔⊔ ΣX is the colimit of any diagram of the form 0←X→0←X→…←X→0.
Step 2. Observe that by sending some X’s to zero in such a diagram, we obtain a map ΣX → ΣX ⊔ ΣX ⊔⊔ ΣX.
Step 3. Observe that this induces maps hom hC(ΣX,Y ) ×× hom hC(ΣX,Y ) → hom hC(ΣX,Y ). Step 4. Observe that any zero morphism ΣX → 0 → Y induces a unit element for this operation.
Step 5. Observe that hom hC(ΣΣX,Y ) admits two multiplications which are compatible. Therefore, by the Eckman-Hilton Lemma they are the same. Consequently, this multiplication is commutative.