# Direct products in subcategories

I have a several categories some of which are subcategories of others. I want to research properties of products in these categories but don't know where to start.

How direct products in a category and its subcategories are related?

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Let $\mathcal{D}$ be a subcategory of $\mathcal{C}$ and let $A$ and $B$ be objects in $\mathcal{D}$. If $A \times_{\mathcal{D}} B$ is the product in $\mathcal{D}$ and $A \times_{\mathcal{C}} B$ is the product in $\mathcal{C}$, then there is a canonical morphism $A \times_{\mathcal{D}} B \to A \times_{\mathcal{C}} B$, but there is not much more we can say than that in general. Here's a somewhat extreme example: let $\mathcal{C}$ be the category of topological spaces and let $\mathcal{D}$ be the frame of open subsets of a topological space $X$; then, $\mathcal{D}$ is a non-full subcategory of $\mathcal{C}$, but the product in $\mathcal{D}$ is the intersection while the product in $\mathcal{C}$ is the cartesian product!
If $\mathcal{D}$ is a full subcategory of $\mathcal{C}$ and $A \times_{\mathcal{C}} B$ is (isomorphic to) an object of $\mathcal{D}$, then it is isomorphic (in $\mathcal{D}$) to $A \times_{\mathcal{D}} B$. In other words, the embedding of a full subcategory reflects products. (In fact, it reflects all limits and colimits.)
If the embedding of $\mathcal{D}$ into $\mathcal{C}$ preserves products, then $A \times_{\mathcal{D}} B$ is isomorphic to $A \times_{\mathcal{C}} B$ in $\mathcal{C}$, provided both of them exist. It is possible for $A \times_{\mathcal{C}} B$ to exist while $A \times_{\mathcal{D}} B$ does not. For example, we could take $\mathcal{D}$ to be the category of fields and $\mathcal{C}$ to be the category of rings.
I do not see how if "$A \times_{\mathcal{C}} B$ is (isomorphic to) an object of $\mathcal{D}$, then it is isomorphic (in $\mathcal{D}$) to $A \times_{\mathcal{D}} B$". I agree that the 2 products are isomorphic in $\mathcal{C}$, but why in $\mathcal{D}$? Consider $\mathcal{C}$ the category with 2 isomorphic objects (1 and 2) and $\mathcal{D}$ the full subcategory with just 1 and its identity. Then the terminal object in $\mathcal{D}$ is isomorphic to 2 in $\mathcal{C}$, but not in $\mathcal{D}$ – magma Feb 9 '14 at 18:02