Equivalence of Subobjects is Equivalence

Theorem

Let $\mathbf C$ be a metacategory.

Let $C$ be an object of $\mathbf C$.

Let $\map {\mathbf {Sub}_{\mathbf C} } C$ be the category of subobjects of $C$.

The relation $\sim$ on $\map {\mathbf {Sub}_{\mathbf C} } C$ defined by:

$m \sim m'$ if and only if $m$ and $m'$ are equivalent

is an equivalence.

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Proof

By Inclusion Relation on Subobjects is Preordering, $\subseteq$ is a preordering.

From Preorder Induces Equivalence, we see that $\sim$ is the equivalence induced by $\subseteq$.

In particular, $\sim$ is an equivalence.

$\blacksquare$