Corollary
Let $\mathcal{C}$ be a category (Definition~Category).
The relation
$$
X \cong Y
$$
defined via isomorphisms (Definition~Isomorphism)
is an equivalence relation on $\operatorname{Ob}(\mathcal{C})$.
Proof
Reflexivity.
For every object $X$, the identity morphism $\operatorname{id}_X$ from
Definition~Category satisfies
$$
\operatorname{id}_X \circ \operatorname{id}_X = \operatorname{id}_X.
$$
Thus $\operatorname{id}_X$ is an isomorphism.
Symmetry.
If $f : X \to Y$ is an isomorphism, Definition~Isomorphism
provides an inverse $f^{-1} : Y \to X$.
Transitivity.
If $f : X \to Y$ and $g : Y \to Z$ are isomorphisms, then
$$
(g \circ f)^{-1} = f^{-1} \circ g^{-1}
$$
by associativity of composition in Definition~Category.
Depends on
Dependency Graph
flowchart LR
classDef current fill:#6366f1,color:#fff,stroke:#4f46e5
nf2c6bdf5["Category"]
n136199b7["Isomorphism"]
n30fdee11["Corollary"]:::current
nf2c6bdf5 --> n30fdee11
n136199b7 --> n30fdee11
click nf2c6bdf5 "../objects/f2c6bdf5.html" "_self"
click n136199b7 "../objects/136199b7.html" "_self"