# Thread: Subobjects in a category

1. ## Subobjects in a category

In a category $\mathcal{C}$ a subobject of an object $A$ is an object $B$ such that there exists a monic $\phi : A \longrightarrow B$. Simple enough.

But I need to show that if $A$ is a subobject of $B$ and $B$ is a subobject of $A$, then $A = B$.

It looks obvious but I think I am missing something.

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