If $∘∘ℎ= ∧ ∘ℎ∘=$ then must $ℎ∘∘=ℎ$?

Question

If not, then What can be said of each $,,ℎ$ and are there any simpy-definable minimal conditions imposable upon one or more of the indexable functions that would ensure this symmetric closure? What about a more generalized linear string form of $_1∘_2∘…∘_=_1 ∧ _2∘…∘_∘_1=_2 ∧ ⋯ ∧ _{−1}∘_∘_1∘…∘_{−2}=_{−1}$ (i.e., does it $⇒ _∘_1∘…∘_{−1}=_$, orand other propert⟮ies⊕y⟯ of $⋀_{i=1}^(_)$ analogous to the short-exact-seq form ⋀(,,ℎ) ? What if every propositional conjunction $∧$ is converted homogeneously to an implication-arrow (one of everyplace either )$⇐,⇒,⇔$; the same uni- or bi- directionality)?