coalgebra homomorphism

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

Definition. Linear map $f:C\to D$ is called coalgebra homomorphism if ${\mathrm{\Delta }}^{\prime }\circ f=(f\otimes f)\circ \mathrm{\Delta }$ and ${\epsilon }^{\prime }\circ f=\epsilon$.

Examples.$1)$ Of course, if $D$ is a subcoalgebra of $C$, then the inclusion $i:D\to C$ is a coalgebra homomorphism. In particular, the identity is a coalgebra homomorphism.

$2)$ If $(C,\mathrm{\Delta },\epsilon )$ is a coalgebra and $I\subseteq C$ is a coideal, then we have canonical coalgebra structur on $C/I$ (please, see this entry (http://planetmath.org/SubcoalgebrasAndCoideals) for more details). Then the projection $\pi :C\to C/I$ is a coalgebra homomorphism. Furthermore, one can show that the canonical coalgebra structure on $C/I$ is a unique coalgebra structure such that $\pi$ is a coalgebra homomorphism.