Categories with $C \times 0 \cong 0$ for all objects $C$ In $Sets$, the initial object is $0 = \emptyset$. We have $C \times \emptyset = \emptyset$ for any set $C$. As there are no maps to the empty set, we don't get much of the universal property of the product. 
I am now wondering what is special about arbitrary categories $\mathscr{C}$ with $C \times 0 \cong 0$ for all objects $C$ of $\mathscr{C}$, where $0$ is the initial object of $\mathscr{C}$. 
From Sheaves in Geometry and Logic by MacLane (p.194), in any topos $\mathscr{C}$, any arrow $k:C \to 0$ is an isomorphism. The universal property of the product would then also not yield very much. 
Are there any categories of higher interest?
 A: Let $\mathscr{C}$ be a category with an initial object $0$, such that the product $C\times 0$ exists for any object $C$ in $\mathscr{C}$. The following are equivalent: 


*

*$C\times 0 \cong 0$ for all objects $C$ in $\mathscr{C}$. 

*Any arrow with codomain $0$ is an isomorphism, i.e. $0$ is a strict initial object. 


For $2\Rightarrow 1$, the projection map $\pi_2\colon C\times 0\to 0$, coming from the definition of the product, is an isomorphism. So $C\times 0\cong 0$. 
For $1\Rightarrow 2$, let $f\colon C\to 0$ be an arrow. Then we have an arrow $(\text{id}_C,f)\colon C\to C\times 0$ and the projection arrow $\pi_1\colon C\times 0\to C$ in the other direction. We have $\pi_1\circ (\text{id}_C,f) = \text{id}_C$, and $(\text{id}_C,f)\circ \pi_1 = \text{id}_{C\times 0}$, since $C\times 0$ is initial. Thus $C\cong C \times 0$, so $C$ is initial. It follows that $f$ is an isomorphism, since the unique arrow between two initial objects is automatically an isomorphism. 

The hypothesis that $C\times 0$ exists for any object $C$ is actually not necessary, since if $0$ is a strict initial object, it follows that $C\times 0$ exists for all $C$. 
Suppose $0$ is strict initial, and let $C$ be an arbitrary object. We'll check that $0$, with the unique projection maps $\pi_1\colon 0\to C$ and $\pi_2\colon 0\to 0$, satisfies the universal property of the product. 
Suppose $X$ is an object and $f\colon X\to C$ and $g\colon X\to 0$ are two arrows. Since $0$ is strict initial, $g$ is an isomorphism, so $X$ is initial. Thus we can define $(f,g) = g\colon X\to 0$, and we have $\pi_1\circ (f,g) = f$ and $\pi_2\circ (f,g) = g$ automatically, since $X$ is initial. 
