If the product of all objects in a finite category exists, is it an initial object? I presume so, but I'm still learning this subject and I can't make a proof go through. Advice welcome. (Not a homework problem, except insofar as I set it for myself.)
By definition, if $x_1, \dots, x_n$ are the objects of the category, their product $\Pi$ is an object equipped with "projections" $p_i : \Pi \to x_i$, such that for every object $x$ of the category and collection of morphisms $f_i : x \to x_i$, there exists a unique morphism $f : x \to \Pi$ such that $p_i \circ f = f_i$.
Consider the category $\mathsf{C}$ with a single object $x$ and two morphisms: $\operatorname{id}_x$ and $f : x \to x$ such that $f \circ f = f$. Then $x$ is of course the product of all the objects in $\mathsf{C}$. But it's not an initial object, because there are two morphisms $x \to x$.