Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows.
$$(f+' g)(x) = f(x)+g(x)$$
Question. The other day, I noticed a suggestive-looking variant of the above definition. However, I'm not sure what to make of this observation. Does it have any particular significance?
The variant.
Let us firstly assign to each $x \in X$ an "evaluation" function $\tilde{x} : Y^X \rightarrow Y$ with defining property $\tilde{x}(f) = f(x).$ This allows $+'$ to be defined as follows, where it is understood that $f$ and $g$ range over all functions $X \rightarrow Y$ and $x$ ranges over every element of $X$.
$$\tilde{x}(f+' g) = \tilde{x}(f)+\tilde{x}(g)$$
In other words, we're defining that $+'$ is the unique binary operation such that for all $x \in X,$ we have that $\tilde{x}$ is a magma homomorphism with source $(Y^X, +')$ and target $(Y,+)$.
Does this final characterization of $+'$ have any particular significance and/or does it "go anywhere"?