Does this thing I'm calling 'the operationalization of $x$' have an accepted name?

Question

Given a set $X$ and an element $x \in X$, we can turn $x$ into a function denoted $\tilde{x}$ as follows: for any set $Y$ and any function $f : X \rightarrow Y$, define $$\tilde{x}(f) = f(x).$$

For my purposes, which are a little too complicated to really describe here, it would be useful to have a name for this process, at least for use in the privacy of my own mind and/or notebook. Furthermore, since this process turns a mere value or element of $X$ into an 'operator' in its own right, I'm tempted to call this process 'operationalization.' So in particular, $\tilde{x}$ is the operationalization of $x$.

Question. Does this thing I'm calling 'the operationalization of $x$' have an accepted name?

Answer 1

In the context of functional analysis, the map $$ \Phi: A \to (\Bbb F)^{\Bbb F^A}\\ \Phi:x \mapsto (f \mapsto f(x)) $$ is called the "evaluation map".

Answer 2

It's called the canonical map into the double dual. It does not have another name that I know of. In a monoidal category with duals (I'm being vague), the uncurried version of it, $\operatorname{ev} : X \otimes X^* \to I$, is called evaluation.

Answer 3

In linear algebra the double-dual comes to mind, but the general term that's most relevant is "evaluation map". One somewhat common notation is $\mathrm{ev}_{x}(f)=f(x)$. Either ev (your ~) is the evaluation map, or each $\mathrm{ev}_{x}$ is the evaluation map at $x$.

Answer 4

Looking at $x \in X$ as a map $x \colon 1 \to X$, the operator $\tilde x$ is precisely the natural transformation given by co-Yoneda's embedding $$ \mathsf{Set}(X,-) \stackrel{\mathsf{Set}(x,-)} \longrightarrow \mathsf{Set}(1,-) \simeq \mathrm{id}_{\mathsf{Set}} $$

So you could call it the restriction along $x$, or precomposition by $x$, and denote it $x^\ast$.