KReiser's comments and FShrike probably already suffice for you and I'm not saying much more but let me write an answer with a little more detail (completely inside the category of $k$-schemes).
Recall us quickly recall the notion of slice categories from ordinary category theory.
Slice categories. Let $\mathscr{C}$ be a category with an object $c \in \mathscr{C}$. Then, $\mathscr{C}_{/c}$ is the slice category (over $c$) where objects are maps $c' \to c$ and maps $(c' \to c) \to (c'' \to c)$ are maps $c' \to c''$ commuting with the structure maps to $c$. Colimits in slice categories are then computed in the underlying category. Written out: If $(c_i \to c)_{i \in I}$ is a diagram of objects in $\mathscr{C}_{/c}$, then one may check that $$\operatorname{colim}_{i \in I} (c_i \to c)_{i \in I} \cong \left(\operatorname{colim}_{i \in I} c_i \to c \right)$$ in $\mathscr{C}_{/c}$ where the map $\operatorname{colim}_{i \in I} c_i \to c$ is induced by the universal property of the colimit.
Now, we may apply it to the concrete setting of $k$-schemes.
Example ($k$-schemes). By definition, the category of $k$-schemes is the slice category $\mathsf{Sch}_{/\operatorname{Spec}{k}}$, and you are given $n$ copies of $\mathrm{id}_{\operatorname{Spec}{k}}:\operatorname{Spec}{k} \to \operatorname{Spec}{k}$ as $k$-schemes. Taking the coproduct inside the category $\mathsf{Sch}_{/\operatorname{Spec}{k}}$ then yields a $k$-scheme $\coprod_{i=1}^n \operatorname{Spec}{k} \to \operatorname{Spec}{k}$ as above with maps induced by the coproduct, where the map is uniquely determined by the $n$-copies of $\operatorname{Spec}{k}$. In particular, this is the preferred $k$-scheme structure. Concretely, it is $\mathrm{id}_{\operatorname{Spec}{k}}$ on each copy of $\operatorname{Spec}{k}$. You may check that this corresponds to the diagonal map $\Delta : k \to k^n$.