Let $V, W_1$ and $W_2$ be vector spaces. A linear map $T : V \to W_1 \times W_2$ is completely determined by its component functions $T_i = \pi_i \circ T$, where $\pi_i$ denotes the standard projection $W_1 \times W_2 \to W_i$. One might even want to write $T = (T_1,T_2)$, or $T = \begin{bmatrix} T_1 \\ T_2 \end{bmatrix}$ in matrix notation.
Dually, a linear map $S : W_1 \times W_2 \to V$ (if I was being picky, I might write $W_1 \oplus W_2$ here instead) is completely determined by the linear maps $S_i=S \circ \iota_i$ where $\iota_i$ denotes the standard inclusion $W_i \to W$. In matrix notation, one might even want to write $S = \begin{bmatrix} S_1 & S_2 \end{bmatrix}$.
Question: Is there a reliable terminology for the maps $S_i$, in the same way that "component function" is reliable terminology for the maps $T_i$?
The best suggestion I can come up with us to call $S_1$ and $S_2$ the factors of $S$. Basically, I have this vague sense that I see the phrases
- "projection onto the $i$th component"
- "inclusion of the $i$th factor"
more often than I see the phrases
- "projection onto the $i$th factor"
- "inclusion of the $i$th component"
but maybe this is just my imagination? What do you think, is "component" more often associated with "product" and "factor" more often associated with "coproduct", in your experience? Or, is it just a wild west and all the terminology is interchangeable? I'd be interested in any opinions/suggestions about what is/isn't good terminology.