I am new to categories. I think the following two questions are silly...
The first question is about definition of category. I feel confused about the definition given in Hungerford's book. In a category, for any pair of objects, e.g. $A$ and $B$, does it necessarily has morphism from $A$ to $B$?
The second question is the meaning of Theorem $8.2$. I don't understand that since the direct product of {$G_i$} is defined to be all functions sending each $i$ into $G_i$ (so in my opinion, direct product is something determined), why the theorem says the product is determined uniquely up to isomorphism?
Thanks!