Coalgebra and coalgebra map over a field

Fix a field $k$ and use algebraic tensor products over $k$ as in Scalars, tensor powers, the empty tensor, opposite algebras and finite sums. A coalgebra is a $k$-vector space $C$ with $k$-linear maps

$$\mathrm{\Delta }:C⟶C\otimes C,\epsilon :C⟶k$$

such that ${\alpha }_{C,C,C}\circ (\mathrm{\Delta }\otimes {\mathrm{i}\mathrm{d}}_C)\circ \mathrm{\Delta }=({\mathrm{i}\mathrm{d}}_C\otimes \mathrm{\Delta })\circ \mathrm{\Delta }$ and ${\lambda }_C\circ (\epsilon \otimes {\mathrm{i}\mathrm{d}}_C)\circ \mathrm{\Delta }={\mathrm{i}\mathrm{d}}_C={\rho }_C\circ ({\mathrm{i}\mathrm{d}}_C\otimes \epsilon )\circ \mathrm{\Delta }\mathrm{.}$ Here $\alpha$ is the tensor associator and ${\lambda }_C,{\rho }_C$ are the tensor unit isomorphisms. They are the maps supplied by Associator naturality, pentagon, unit triangle and symmetry hexagons on elementary tensors and Scalars, tensor powers, the empty tensor, opposite algebras and finite sums. Thus the coassociativity equation has common target $C\otimes (C\otimes C)$; no unmentioned equality of differently parenthesized tensor spaces is assumed. The tensor products are algebraic, so each tensor is a finite sum of elementary tensors.

If $C,D$ are coalgebras, a coalgebra map $f:C\to D$ is a $k$-linear map satisfying

$${\mathrm{\Delta }}_D\circ f=(f\otimes f)\circ {\mathrm{\Delta }}_C,{\epsilon }_D\circ f={\epsilon }_C\mathrm{.}$$

The equations say exactly that $f$ preserves comultiplication and counit.

The defining equations are properties; they do not assert existence for every vector space.