Show that an equivalence of categories sends products to products and coproducts to coproducts. That is, if $X_i$ are a family of objects in $\mathcal{C}$ with coproduct $X$ then $F(X)$ is the coproduct of $F(X_i)$ in $\mathcal{D}$, and similarly for products.
Hint on how to get started?
Let me first say that I'm confused what really needs to be shown. a) Manipulate $F(X_i)$ etc directly, and construct the "connecting morphism" (I think you understand what I mean by that) somehow. b) Start with a collection $(Y_i)$ in $\mathcal{D}$ and any object $D$ equipped with morphisms $g_i : Y_i \rightarrow D$, and show that there exists a unique morphism $\theta':F(X) \rightarrow D$ such that $g_i = \theta' F(\alpha_i)$ (question is, can I really assume that such a $D$ and $g_i$ exist?). Anyway, here's what I've tried (full of holes)
Let $D$ be an object in $\mathcal{D}$ equipped with morphisms $g_i : Y_i \rightarrow D$. Let $F:\mathcal{C} \rightarrow \mathcal{D}$, $G:\mathcal{D} \rightarrow \mathcal{C}$ and put $X_i=G(Y_i)$, $f_i=G(g_i)$, $C=G(D)$. Then $(C,f_i)$ is such that there exists a unique morphism $\theta:X \rightarrow C$ such that $f_i=\theta \alpha_i$, where $\alpha_i : X_i \rightarrow X$. Let $\varepsilon : FG \rightarrow \mathrm{id}_{\mathcal{D}}$ be a natural isomorphism. This means we have isomorphisms $\varepsilon_{Y_i}:F(X_i)=FG(Y_i) \rightarrow Y_i$ and $\varepsilon_D:F(C)=FG(D) \rightarrow D$ such that $\varepsilon_D FG(g_i)=g_i \varepsilon_{Y_i}$. LHS equals $\varepsilon_D F(f_i)=\varepsilon_D F(\theta \alpha_i)=\varepsilon_D F(\theta) F(\alpha_i)$.
But yeah, this doesn't achieve much. All my ideas are along these lines but I can't get anything useful out of it. Many thanks for advice on this.