The two morphisms
λ_ (𝟙_ C) and
ρ_ (𝟙_ C) from
𝟙_ C ⊗ 𝟙_ C to
𝟙_ C are equal.
This is suprisingly difficult to prove directly from the usual axioms for a monoidal category!
This proof follows the diagram given at https://people.math.osu.edu/penneys.2/QS2019/VicaryHandout.pdf
It should be a consequence of the coherence theorem for monoidal categories (although quite possibly it is a necessary building block of any proof).