Definition 1

Tag 08HO, permanent. Cite it as [Bourbaki, Tag 08HO].

Functions of a Real Variable, chapter VI, § 1. GENERALIZED TAYLOR EXPANSIONS

One says that an operator U on K[X] is a composition operator if it commutes with the translation operator, that is, if $U_X T_Y = T_Y U$ .

In other words, if $f$ is an arbitrary polynomial in K[X], and if $g = U(f)$ , one must have $g(X + Y) = U_X(f(X + Y))$ .

It follows immediately from this definition that for every polynomial $f(X) \in K[X]$ one has, in the above notation,

U_X(f(X + Y)) = U_Y(f(X + Y)).

(1)

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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