Vtyjkyuk

I am studying the Geometry book by Michèle Audin, let me recall her definition of an affine space.

A set $mathcalE$ is endowed with the structure of an affine space by the data of a vector space $E$ and a mapping $Theta$ that associates a vector of $E$ with any ordered pair of points in $mathcalE$,
$$
mathcalEtimesmathcalEto E \
(A,B)mapsto overrightarrowAB
$$
such that:

• for any point $A$ of $mathcalE$, the partial map $Theta_A:Bmapsto overrightarrowAB$ is a bijection from $mathcalE$ to $E$;


• for all points $A,B$ and $C$ in $mathcalE$, we have $overrightarrowAB=overrightarrowAC+overrightarrowCB$ (Chasles' relation).

Now her definition of an affine map.

Let $mathcalE$ and $mathcalF$ be two affine spaces directed
respectively by $E$ and $F$. A mapping
$phi:mathcalEtomathcalF$ is said to be affine if there
exists a point $O$ in $mathcalE$ and a linear mapping $f:Eto F$
such that $$ forall MinmathcalE, qquad
f(overrightarrowOM)=overrightarrowphi(O)phi(M).
$$

My question is the following. Could it be possible to define an affine map as a map $phi:mathcalEtomathcalF$ such that there exists a point $OinmathcalE$ and a linear map $f:Eto F$, which has the following compatibility condition with the map $Theta$:
$$
f(Theta_O(M)) = Theta_phi(O)(phi(M)) qquad forall MinmathcalE
$$
?

edit: after some thinking, I would also rephrase my question as: is it true that a map $phi:mathcalEtomathcalF$ is affine if and only if there exists a linear map $f:Eto F$ such that the following diagram:
$$
requireAMScd
beginCD
E @>f>> F \
@ATheta_OAA @ATheta_phiOAA \
mathcalE @>phi>> mathcalF
endCD
$$
commutes?