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Suppose $C : [0,1]^2 \mapsto [0,1]$ is a copula function (i.e. a function whose range is the bivariate CDF and whose arguments are both the marginal univariate CDFs). Define $a \in [0,1]$ where the horizontal section of $C$ at $a$ is the function from $[0,1]$ to $[0,1]$ given by $t \mapsto C(t,a)$.

My question is what is this "$t \mapsto C(t,a)$" ??

$t$ hasn't been defined before and since the copula's domain is $[0,1] \times [0,1]$ I'm guessing that it's just some constant in $[0,1]$.

But since it's a constant why are they using the mapping notation?

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This is the function $u:[0,1]\to\mathbb R$ such that $u(t)=C(t,a)$ for every $t$ in $[0,1]$. One can use the notation $u=C(\cdot,a)$.

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