In mainstream notation for composition, by $g \circ f$ we mean $\operatorname{dom} \left(g \circ f \right) = \operatorname{dom}\left(f\right)$ and $\operatorname{cod} \left(g \circ f \right) = \operatorname{cod}\left(g\right)$
However, when watching a story about categories, I prefer seeing paths going from domains to codomins, i.e. like a linear narrative. But when writing, I have to take care of a kind of flashback, when writing. Also, I know that many accomplished mathematicians do use the reversed notation of the mainstream.
I would like to know what are the pros and cons of these two notations and how harmful can be, getting used to a non-mainstream notation?