Let $\mathcal{F} : A \to F$ be a function, and $ f = \mathcal{F}(a)$.
$\big($with $a\in A$ and $f\in F$$\big)$.
Now, let $f : X \to Y$ itself be another function, and $y = f(x)$.
$\big($with $x\in X$ and $y\in Y$$\big)$.
What is the adequate notation to write:
$y = f(x) \overset{?}{=} \mathcal{F}(a)(x)$.
Thanks for your answers.
NS
There are three commonly-used notations in math: $$ \mathcal{F}(a) = f \ \ \text{ where } \ \ f(x) = a+x \\ \mathcal{F}(a)(x) = a+x \\ \mathcal{F}(a) = x \mapsto a+x $$ and one in computer-science (I give two examples, but it is the same notation): $$\mathcal{F}(a) = \lambda x.\ a+x\\ \mathcal{F} = \lambda a.\ \lambda x.\ a+x$$ and you can note the corresponding types as, for example, $\mathcal{F}: \mathbb{R} \to \mathbb{R}^{\mathbb{R}}$ or $\mathcal{F}: \mathbb{R} \to (\mathbb{R} \to \mathbb{R})$.
Each of these define $\mathcal{F}$ where $\mathcal{F}(0)$ is the identity function, and $\mathcal{F}(1)$ is the add-one function on $\mathbb{R}$.
I hope this helps $\ddot\smile$