
Subobjects in a category
In a category $\displaystyle \mathcal{C}$ a subobject of an object $\displaystyle A$ is an object $\displaystyle B$ such that there exists a monic $\displaystyle \phi : A \longrightarrow B$. Simple enough.
But I need to show that if $\displaystyle A$ is a subobject of $\displaystyle B$ and $\displaystyle B$ is a subobject of $\displaystyle A$, then $\displaystyle A = B$.
It looks obvious but I think I am missing something. (Wondering)

Ah, If $\displaystyle \theta : B \longrightarrow A$ is the other monic then their composition is monic, but their compositions are identity morphisms for A and B; $\displaystyle \theta \circ \phi = 1_A$ and $\displaystyle \phi \circ \theta = 1_B$. Hence they are isomorphic! Or is there a strict equality? (Thinking) help