Suppose we have objects $X,Y$, and product $X\times Y$. The definition of the product says that for any other object $Z$, and pair of morphisms $f:Z\to X, g:Z\to Y$, there is a corresponding product morphism $f\circ g$.
But what if there are no morphisms at all from $Z$ to $Y$ say, but there are many morphisms from $Z$ to $X$? Do we know anything about the morphisms from $Z$ to $X\times Y$? Can there be any morphisms? do there have to be any morphisms?
There are no morphism from $Z$ to $X\times Y$ since for such a morphism $p_Y\circ f:Z\rightarrow Y$ and $p_X\circ f:Z\rightarrow X$ are defined where $p_X:X\times Y\rightarrow X$ is the projection.