Proposition
Uniqueness of Inverses
Let $\mathcal{C}$ be a category (Definition~Category).
If a morphism $f : X \to Y$ is an isomorphism (Definition~Isomorphism),
then its inverse is unique.
Proof
Suppose
$$
g,h : Y \to X
$$
are both inverses of $f$. Then
$$
g \circ f = \operatorname{id}_X,
\qquad
f \circ g = \operatorname{id}_Y,
$$
and
$$
h \circ f = \operatorname{id}_X,
\qquad
f \circ h = \operatorname{id}_Y.
$$
Using associativity of composition from Definition~Category we obtain
$$
g
=
g \circ \operatorname{id}_Y
=
g \circ (f \circ h)
=
(g \circ f) \circ h
=
\operatorname{id}_X \circ h
=
h.
$$
Thus the inverse is unique.
Depends on
Dependency Graph
flowchart LR
classDef current fill:#6366f1,color:#fff,stroke:#4f46e5
nf2c6bdf5["Category"]
n136199b7["Isomorphism"]
n45ea5662["Uniqueness of Inverses"]:::current
nf2c6bdf5 --> n45ea5662
n136199b7 --> n45ea5662
click nf2c6bdf5 "../objects/f2c6bdf5.html" "_self"
click n136199b7 "../objects/136199b7.html" "_self"