coalgebra isomorphisms and isomorphic coalgebras

Let $(C,\mathrm{\Delta },\epsilon )$ and $(D,{\mathrm{\Delta }}^{\prime },{\epsilon }^{\prime })$ be coalgebras.

Definition. We will say that coalgebra homomorphism $f:C\to D$ is a coalgebra isomorphism, if there exists a coalgebra homomorphism $g:D\to C$ such that $f\circ g={\mathrm{id}}_D$ and $g\circ f={\mathrm{id}}_C$.

Remark. Of course every coalgebra isomorphism is a linear isomorphism, thus it is ,,one-to-one” and ,,onto”. One can show that the converse also holds, i.e. if $f:C\to D$ is a coalgebra homomorphism such that $f$ is ,,one-to-one” and ,,onto”, then $f$ is a coalgebra isomorphism.

Definition. We will say that coalgebras $(C,\mathrm{\Delta },\epsilon )$ and $(D,{\mathrm{\Delta }}^{\prime },{\epsilon }^{\prime })$ are isomorphic if there exists coalgebra isomorphism $f:C\to D$. In this case we often write $(C,\mathrm{\Delta },\epsilon )\simeq (D,{\mathrm{\Delta }}^{\prime },{\epsilon }^{\prime })$ or simply $C\simeq D$ if structure maps are known from the context.

Remarks. Of course the relation ,,$\simeq$” is an equivalence relation. Furthermore, (from the coalgebraic point of view) isomorphic coalgebras are the same, i.e. they share all coalgebraic properties.