A question related to Continuing direct product on a subcategory.
Let $F$ is a full subcategory of a category $G$.
I denote $\operatorname{Ob}X$ the set of objects of a category $X$.
Is it possible the following?
- There are binary direct products $\times_F$ and $\times_G$ in these categories.
- $A\times_G B\in\operatorname{Ob}F$ if $A, B\in\operatorname{Ob}F$.
- $A\times_G B$ is non-isomorphic to $A\times_F B$ for some objects $A$, $B$ of $F$.
We probably should also assume that the product morphisms are the same for $F$ and $G$ (provided that the morphisms are morphisms of $F$).
No. Under the given hypotheses, $A \times_G B \cong A \times_F B$ naturally. To see this it suffices to observe that they represent the same functor when restricted to $F$.