In the wiki page:https://en.wikipedia.org/wiki/Cofree_coalgebra
They discuss two coalgebra structures on $T(V)$
I dropped the tensor between $v_1\otimes \cdots\otimes v_n$, the two coproducts: $$\Delta(v_1\cdots v_n) = \sum_{p=0}^n v_0\cdots v_p \otimes v_{p+1}\cdots v_{n+1}$$ $$\Delta(v_1\cdots v_n) =1\otimes v_1\cdots v_n+\sum_{p=1}^{n-1} \sum_\sigma v_{\sigma(1)} \cdots v_{\sigma(p)}\otimes v_{\sigma(p+1)}\cdots v_{\sigma(n)}+v_1\cdots v_n\otimes 1$$ Where $\sigma(1)<\sigma(2)<\cdots < \sigma (p) $ and $\sigma(p+1)<\cdots < \sigma(n)$
I am trying to solve one of Kessel's book (Quantum groups) problems that says:
Blockquote: (Chapter 3, exercise 3) Show that the canonical isomorphism $V^{\otimes n}\otimes V^{\otimes m} \cong V^{\otimes (n+m)}$ endow $T(V)$ with a coalgebra structure
My first thought was $\Delta x = x\otimes x$ i.e. $$\Delta (v_1\cdots v_n) = v_1\cdots v_n \otimes v_1\cdots v_n$$ And it is a co-associative $$\Delta \otimes I \circ \Delta(x)= (x\otimes x)\otimes x=x\otimes x \otimes x=x\otimes (x \otimes x)=I \otimes \Delta \circ \Delta (x)$$ If we define $\varepsilon (x) = 0$ Then $$(I\otimes \varepsilon )\circ \Delta x = x\otimes 0 = 0= 0\otimes x = \varepsilon \otimes I \circ \Delta (x)$$
In the book, they did not discuss how to come up with the co-algebra. All examples seem like they choose any definition that works with the co-associativity and co-unity.