types - Is there a way to demonstrate uniqueness of false-elim -

i can't remember if i've read somewhere, tempting assume ⊥ initial object. must possible construct proofs based on uniqueness of ⊥-elim arrows.

like this:

false-elim : forall {a : set} -> false -> false-elim ()  false-iso : forall {a b : set} -> (g : -> false)                                -> (f : -> b) -> f == (f o false-elim o g) 

that is, if there arrow ⊥, isomorphic ⊥. ok, if assumption (a -> ⊥) isomorphism wrong, @ least must possible show uniqueness of ⊥-elim:

false-elim-uniq : forall {a b : set} -> (f : -> b)                                      -> false-elim == (f o false-elim) 

but not obvious, too. so, ⊥-elim meant unique in (flavour of) intuitionistic type theory (agda based on)?

it possible construct proof if element of can constructed:

false-iso : forall {a b : set} -> (g : -> false)                                -> (f : -> b) -> -> f == (f o false-elim o g) 

but that's not quite same statement. can prove homotopy of said functions (and show ≅ ⊥).

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given thought, guess can narrow down problem to: if can show homotopy f ~ (f o false-elim o g), universal property of false-elim like?