I have read that, given two input elements $(a,b)$, and given a binary operation $*$, a map $F$ is said to be operation-preserving if $F(a*b) = F(a)*F(b)$. For example, linear maps always preserve the operation of vector addition (as well as vector multiplication): $F(a+b) = F(a)+F(b)$. (I'm aware that in general there should be two operations $($#$,*)$ such that $F(a$#$b) = F(a)*(b)$, here just for simplicity I'm considering the case in which # = $*$, as in the vector addition example).
At the same time i have read here that:
If a map $T:V \to V$ preserves $\langle\cdot,\cdot \rangle$, then $\langle Tv,Tw \rangle = \langle v,w \rangle$ for all $v,w \in V$.
So when the operation $*$ is the inner product $\langle\cdot,\cdot \rangle$, the "operation preserving"-propriety of a map $F$ can be written as $(a*b) = F(a)*F(b)$, and not anymore as $F(a*b) = F(a)*F(b)$ as stated in the beginning. If it could be written as $F(a*b) = F(a)*F(b)$, then we would have: $\langle Tv,Tw \rangle = T\langle v,w \rangle$.
So my question is: in general, when can we write the operation-preserving propriety of a map as $(a*b) = F(a)*F(b)$ (e.g: operation of inner product) and when can we write it as $F(a*b) = F(a)*F(b)$ (e.g.: operation of vector addition)?
Please note that i'm interested in a general answer, not specific to the inner product / vector addition examples.
