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Let $$f : X_0 \rightarrow Y_0, \;\;\; g:X_1 \rightarrow Y_1$$ and define that the "direct product" of $f$ and $g$ is a map $$f \otimes g : (X_0 \times X_1) \rightarrow (Y_0 \times Y_1), \mbox{ such that } (f \otimes g)(x_0,x_1)=(f(x_0),g(x_1)).$$

Question 1. What is the standard terminology/notation for this concept?

Now given a function $h$, it's possible that $h$ has the property that there exist $f$ and $g$ such that $$h = f \otimes g$$ Question 2. What is the name of this property?

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What context are you working in? In additive categories I would denote this $\begin{pmatrix}f&0\\0&g\end{pmatrix}$ – Julian Kuelshammer Sep 6 '12 at 11:31
I know this as the "product of two functions", since it is just the result of applying the product functor $- \times -: Set \times Set \to Set$ to a pair of functions/morphisms. – Todd Trimble Sep 6 '12 at 11:49
This condition comes up often enough for it really to deserve a name. I never found a standard one and so have used the phrase "$h$ splits multiplicatively" in publications. Despite its apparent simplicity it is useful to have criteria which one can use to determine if $h$ has this property. Two very straightforward ones which are valid under suitable assumptions are: a) for all pairs $x_0$, $x_1$ and $y_0$, $y_1$ we have: $h(x_0,x_0)h(x_1,y_1)=h(x_0,y_1)h(x_1,y_0)$. b) $h h_{xy} = h_x h_y$ (this for smooth functions). – jbc Sep 6 '12 at 13:22

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