Equivalence of Subobjects is Equivalence

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Theorem

Let $\mathbf C$ be a metacategory.

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

Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$.


The relation $\sim$ on $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ defined by:

$m \sim m'$ iff $m$ and $m'$ are equivalent

is an equivalence.


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.

$\blacksquare$