Full embeddings of categories

Question

By the book “a course in homological algebra”, by Hilton and Stammbach, on pages 74, 75, for abelian categories we have a full embedding theorem which as I understood, we can use this theorem to compute the limits and colimits of finite diagrams but not the infinite ones. My first question is that why it is true. More generally, if we have a full embedding of a category in another category, let’s say, we have two categories $C$ and $D$ and a functor $F:C\longrightarrow D$ which fully embedded $C$ in $D$, then is it true that the finite limits and finite colimits in $F(C)$, $F(C)$ as a category, are the same as the finite limits and finite colimits of our main diagram in $F(C)$ but as a diagram in $D$? My second question is that why it is not true for the infinite case.