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Let $(C,\Delta,\varepsilon)$ and $(D,\Delta',\varepsilon')$ be coalgebras.
Definition. Linear map $f:C\to D$ is called coalgebra homomorphism if $\Delta'\circ f=(f\otimes f)\circ\Delta$ and $\varepsilon'\circ f=\varepsilon$
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 $(C,\Delta,\varepsilon)$ is a coalgebra and $I\subseteq C$ is a coideal, then we have canonical coalgebra structur on $C/I$ (please, see this entry 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.
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