It is well known that a quadratic algebra $A(V,R)$ is the quotient of the free associative algebra $T(V)$ over a vector space $V$ by the two-sided ideal $(R)$ generated by $R\subseteq V^{\otimes 2}$, i.e., we have the short exact sequence $0 \to R \to T(V) \to A \to 0$, and the set $V$ is called generators and the set $R$ is called relations.
Dually, one can define a notation of quadratic coalgebra $C(V,R)$ as a sub-coalgebra of the cofree coassociative coalgebra $T^c(V)$ such that the following short sequence is exact: $0 \to C \to T^c(V) \to V^{\otimes 2}/R \to 0$. They say that $C(V,R)$ is cogenerated by $V$ with corelations $R$.
The coalgebra $C(V,R)$ is weighted graded. Explicitly it is given by: $$ C = \bigoplus_{n \in \mathbb{N}} C^{(n)} = \Bbbk \oplus V \oplus R \oplus \cdots \oplus \left( \bigcap_{p+q=n-2} V^{\otimes p} \otimes R \otimes V^{\otimes q} \right) \oplus \cdots \qquad (*) $$
For example, let $R = \{x\otimes x, y \otimes y\}$, then by the previous formula we obtain $C = \{ \alpha, x, y, x^{\otimes n}, y^{\otimes n},\, \alpha \in \Bbbk, n \in \mathbb{N} \}$. Indeed, we have $T^c(V)/C \cong V^{\otimes 2}/R = \mathrm{Span}(x\otimes y, y \otimes x).$
Next, let $R = \{x\otimes y - y\otimes x\} \subset V\otimes V$. And I have troubles to calculate $C$ by (*). Let us consider $V\otimes R \cap R \otimes V$. As I guess it is zero space. Let $v = \alpha x + \beta y$ and $u = \alpha'x + \beta' y$ be vectors of $V$ with $\alpha, \alpha', \beta, \beta' \in \Bbbk$. To find $V\otimes R \cap R \otimes V$ we have to consider to solve an equation $v \otimes (x\otimes y - y \otimes x) = (x\otimes y - y \otimes x) \otimes u $. But it is easy to see that $v = u=0$. Thus $V\otimes R \cap R \otimes V=0$.
It seems that $C = \{\alpha, x, y, x\otimes y - y\otimes x,\, \alpha \in \Bbbk\}$, but, on the other hand $V^{\otimes 2}/R = \mathrm{Span}(x\otimes x, y \otimes y, x\otimes y)$, and it must be $V^{\otimes 2}/R \cong T^c(V)/C$, and I don't understand how it is possible. I also cannot understand how to describe a map $T^c(V) \to V^{\otimes 2}/R$.
So, how to describe this coalgebra? Help me, please.